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Today, we're discussing how to calculate wave speed. Can anyone tell me the formula for wave speed?
Isn't it related to frequency and wavelength?
Exactly! The wave speed v is given by the formula v = fλ, where f is the frequency and λ is the wavelength. Can anyone explain why this relation holds true?
Because a wave travels a certain distance, which in this case is the wavelength, in a certain time based on its frequency!
Correct! Now let's remember this: to find the speed of a wave, think of it as the product of how many cycles pass per second times the distance of one cycle. That’s a useful memory aid!
Can we see an example?
Sure! If we have a wave with a frequency of 5 Hz and a wavelength of 3 m, what’s the speed?
I think it's 15 m/s!
Exactly! Great job, everyone. Remember, when we write the formula v = fλ, it’s always good to connect back to how each component affects wave motion.
Let’s talk about the types of waves we learned about. Can someone explain the difference between transverse and longitudinal waves?
Transverse waves are when the particles move perpendicular to the direction of the wave.
Correct! What about longitudinal waves?
In longitudinal waves, the particles move parallel to the direction of wave motion.
Exactly! A simple way to remember this is 'T' for 'transverse' means 'up and down', while 'L' for 'longitudinal' can remember as 'line' following the wave, aligning in the same direction.
Can you give an example of each?
Sure! Water waves are often seen as transverse, while sound waves in the air are longitudinal. Now, let's look for examples in everyday scenarios to reinforce our understanding!
Let’s start applying what we learned. If an ultrasonic sound wave travels through air with a frequency of 1000 kHz, what can we deduce about its behavior in water?
The wavelength will be different based on the speed of sound in air and water.
Exactly! And since sound travels faster in water, we would calculate the wavelength using v = fλ. What do we know about the speed of sound in both mediums?
It’s approximately 343 m/s in air and around 1486 m/s in water.
Well done! Now, if you compare and find the wavelength in both media, it will help you visualize how sound behaves differently in various conditions.
This shows how important the medium is in wave behavior!
Precisely! Understanding these connections is key to mastering wave concepts.
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The exercises are divided into easy, medium, and hard levels, each designed to test the understanding of key concepts covered in the chapter on waves, including wave properties, behaviors, and equations.
In this section, you will find a series of exercises designed to reinforce your understanding of the concepts presented in the chapter on waves. These exercises vary in difficulty, providing a structured approach to assess your grasp of topics such as wave speed, types of waves, superposition principle, and more. Each exercise encourages critical thinking and application of the learned concepts in practical scenarios related to wave behavior in different media.
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14.1 A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
In this exercise, we're asked to calculate how long it takes for a disturbance to travel along a string. We can use the wave speed formula for a stretched string, which is determined by the tension in the string and its mass per unit length. The speed of a wave on a string is given by the formula v = √(T/μ), where T is the tension and μ is the linear mass density. First, we calculate the linear mass density (μ) by dividing the mass of the string by its length. Then we find the speed (v) using the tension provided. The time taken for the disturbance to travel the entire length of the string can be calculated using the formula time = distance/speed.
Imagine plucking a guitar string. When you pluck it, a wave travels down the string, and we can understand this exercise as calculating how quickly you would hear the sound of your pluck from the other end of the string based on the characteristics of the string itself.
14.2 A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m s–1 ? (g = 9.8 m s–2)
In this exercise, we are to calculate the total time it takes from when a stone is dropped to when its splash is heard. First, we calculate the time it takes for the stone to fall to the water using the equation of motion: time taken to fall (t_fall) is given by t_fall = √(2h/g), where h is the height of the tower and g is the acceleration due to gravity. After the stone hits the water, the sound of the splash travels upwards to the top of the tower. The time taken for sound to travel back up (t_sound) is given by t_sound = distance/speed of sound. The total time is the sum of these two times.
Think of a scenario in a park where you drop a ball into a pond. You will see it hit the water first. After a moment, you’ll hear the splash. This exercise helps us quantify how long that moment is when dropping a stone from a tall tower, simulating a real-life splash scenario.
14.3 A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 °C = 343 m s–1?
This exercise involves relating the speed of a transverse wave in a wire to the tension and the mass density of the wire. We start by calculating the linear mass density of the wire, which is mass divided by length. Next, we use the wave speed formula for a string, v = √(T/μ), where v is the desired speed (343 m/s), μ is the linear mass density we calculated, and T is the unknown tension. Solving for T will give us the required tension to achieve the desired wave speed.
Consider tightening a rope to create a musical tension. This exercise relates to adjusting the tension of a wire, similar to tuning a guitar string to a particular frequency by increasing the tension to achieve the right pitch.
14.4 Use the formula v = P/ρ to explain why the speed of sound in air (a) is independent of pressure, (b) increases with temperature, (c) increases with humidity.
This exercise requires an understanding of how pressure, temperature, and humidity affect sound speed in fresh air. For part (a), the speed of sound in an ideal gas depends on its bulk modulus and density and is independent of pressure at constant temperature. Part (b) states that increasing temperature reduces air density, increasing sound speed, while for part (c), higher humidity implies that water vapor, being lighter than nitrogen and oxygen, reduces the overall density of air, leading to an increased sound speed. Each condition alters how quickly sound waves propagate.
Think about how sound travels faster on a hot summer day than on a cold winter day. Similarly, near a body of water, you might notice how sound seems to travel differently when the air is humid versus dry. This exercise illustrates those differences in terms of physics using fundamental principles.
14.5 You have learned that a travelling wave in one dimension is represented by a function y = f(x, t) where x and t must appear in the combination x – vt or x + vt, i.e., y = f(x ± vt). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave: (a) (x – vt)² (b) log[(x + vt)/x₀] (c) 1/(x + vt)
This exercise challenges the concept of wave representation in mathematics. It asks to determine if certain mathematical expressions can represent travelling waves. To do this, one must identify if they conform to the appropriate mathematical form of a travelling wave, namely f(x ± vt). Functions like (x - vt)² and log[(x + vt)/x₀] do not depict a simple wave function because their outputs do not represent oscillations or sinusoidal behavior. Meanwhile, 1/(x + vt) also does not fit the characteristics of wave propagation because it suggests a behavior that fluctuates inversely instead of oscillating in a wave pattern.
Think of trying to visualize different types of curves. Just as not all curves represent a wave moving through the air, not all mathematical functions do. Just like observing how sound waves pulse, the challenge here is recognizing those functions that don't authentically depict a wave's essence.
14.6 A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m s–1 and in water 1486 m s–1.
This exercise deals with ultrasound propagation in different media. For part (a), we can find the wavelength of the ultrasonic sound in air using the formula λ = v/f, where v is the speed of sound in air, and f is the frequency. Part (b) requires calculating the wavelength of the sound in water using the same formula but substituting in the speed of sound in water. This exercise illustrates how sound travels differently depending on the medium, affecting wavelength while frequency remains constant.
Imagine a dolphin, which uses similar high-frequency sounds to navigate and hunt underwater. This exercise highlights how sounds behave differently above and below the water, just like how a bat's echolocation changes when the sound hits the water.
14.7 A hospital uses an ultrasonic scanner to locate tumours in tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s–1? The operating frequency of the scanner is 4.2 MHz.
To determine the wavelength of the sound waves in tissue, we’ll use the formula λ = v/f. In this case, v is 1.7 km/s (which we convert to m/s for consistency), and f is the frequency given in megahertz (4.2 MHz, converted into hertz). By calculating λ, we establish how ultrasound interacts within tissue, critical for medical diagnostics.
Consider how doctors visualize the body with ultrasound imaging, using sound waves to see inside without surgery. This exercise quantifies the wavelength used for imaging, helping to understand its diagnostic role.
14.8 A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin(36 t + 0.018 x + π/4) where x and y are in cm and t in s. The positive direction of x is from left to right. (a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation? (b) What are its amplitude and frequency? (c) What is the initial phase at the origin? (d) What is the least distance between two successive crests in the wave?
First, we analyze the wave function's form to identify traveling versus stationary characteristics. The standard form for a traveling wave is identifiable by its x and t dependencies, revealing its speed and direction. Amplitude indicates maximum displacement in relation to equilibriums, and frequency describes oscillation rate. The initial phase shows displacement value starting at time zero. Lastly, the distance between crests, or wavelength, is derived from the wave number component within the sine function.
Think of a ripple travelling along a surface of a pond when a stone is dropped; this exercise similarly dissects how waves move and behave, analyzing their properties similar to observations in nature.
14.9 For the wave described in Exercise 14.8, plot the displacement (y) versus (t) graphs for x = 0, 2, and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in a travelling wave differ from one point to another: amplitude, frequency or phase?
This exercise involves graphing the displacement over time at specific points along the wave. By plugging in values of x into the equation gives y(t) for different locations. From these results, the shape of the graphs can be analyzed. Importantly, this allows us to distinguish between oscillations at different points; while frequency may remain constant, phase and potentially amplitude can differ.
Imagine different spots along a wave crashing on a shore; while they all experience the same ocean wave frequency, each point's timing and energy may vary slightly, which provides richness to the wave's action. This exercise mimics that observation across a wave's points.
14.10 For the travelling harmonic wave y(x, t) = 2.0 cos 2π(10t - 0.0080x + 0.35) where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (a) 4 m, (b) 0.5 m, (c) λ/2, (d) 3λ/4.
In this exercise, we are going to explore the concept of phase difference across varying distances. Using the relationship of the wave function and calculating changes over defined distances into phase terms allows us to reveal how displacement varies over distance. Phase differences help visualize the point's movement between crests and troughs, which can then be defined using the wavelength in relation to the described distances.
Think of dancers in a performance; even if two dancers start from the same point, they can vary in phase or sync as they navigate the stage. By calculating the phase difference, this exercise harnesses similar dynamics, mirroring how wave oscillations interact along their path.
14.11 The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin(2 3πx) cos(120πt) where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 × 10–2 kg. Answer the following : (a) Does the function represent a travelling wave or a stationary wave? (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave? (c) Determine the tension in the string.
This problem explores the nature of standing waves created along a string. The wave function provided contains both sine and cosine components, indicative of superposition of two waves traveling in opposite directions. From their mathematical forms, we can extract wavelength, frequency, and interpret what the tension in the string can produce such wave action. Relationships derived from the wave characteristics leading back to physical properties allow us to gain insights into real wave mechanics.
Picture a taut string during a performance: as musicians play, the vibrations travel up and down, creating visible undulations. Breaking down that wave action into motions can help us understand how forces within the string create beautiful music by harmonizing vibrations.
14.12 (i) For the wave on a string described in Exercise 14.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?
In this exercise, we delve deeper into understanding oscillations along the entire length of the string. Given that waves can differ in amplitude and phase even while sharing the same frequency, students must analyze a specific point's characteristics based on distance from a node or antinode along the string. We derive amplitude by evaluating the original wave function and substituting the specific x-value to find the characteristics distinctively at that position.
Think about how the sounds vary as you move away from a concert speaker: while the frequency remains consistent, the sound's strength (amplitude) changes as does its clarity (phase). This exercise parallels that observation by examining vibrational patterns along the string.
14.13 Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (a) y = 2 cos (3 x) sin (10 t) (b) y = 2 (x – vt) (c) y = 3 sin (5 x – 0.5 t) + 4 cos (5 x – 0.5 t)
This exercise tests the understanding of different forms of wave representations, distinguishing between traveling and stationary waves. Students must dissect functions by evaluating their mathematical structures, comparing against criteria for wave solutions, examining if they can express movement effectively. Each mathematical form can project how oscillations occur and whether or not they accurately reflect a traveling pattern.
Like painting a moving wave on canvas, this exercise encourages clarity in representation: observing which patterns truly illustrate motion and which might not capture its dynamic essence accurately can guide us in understanding nature's rhythmical dance and flow.
14.14 A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10–2 kg and its linear mass density is 4.0 × 10–2 kg m–1. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?
In this exercise, we will determine how the properties of a wire relate to wave mechanics. (a) The speed of the transverse wave can be calculated using the relationship v = √(T/μ). We can thus derive the speed knowing the linear mass density and relating it back to the frequency. (b) Since we've derived the wave speed, we can then compute the underlying tension using T = μv²; overall, this exercise showcases the interconnectedness of frequency, tension, and vibration on strings.
Consider an orchestra tuning their instruments before a performance; they adjust string tension to ensure the right notes are played. This exercise embodies that principle—calculating how a precise tension creates harmony within sound waves produced through the oscillation of strings.
14.15 A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
In this exercise, we will measure how sound behaves in a tube with one end open, observing resonance positions. Knowing tube lengths that resonate allows the calculation of wavelengths, which, in conjunction with the frequency, lets us estimate the speed of sound in air using the relationship v = fλ, where f is frequency and λ is wavelength. This embodies real-world sound behavior, particularly in instruments and acoustics.
Imagine blowing into a straw; your pitch changes based on how much of the straw is submerged in liquid. This exercise quantifies that resonance concept, demonstrating how tube dimensions and sound interact, similar to playing different notes by altering straw lengths in experiments.
14.16 A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
This exercise explores the concept of wave propagation through a solid medium, a steel rod in this case. We can determine the speed of sound using the relationship for the fundamental frequency v = nλ/2L, where n is the harmonic number (1 for fundamental mode), L is the length of the string, and λ can be derived from the relationship with frequency and speed. Essentially, this highlights how sound travels efficiently through solid materials.
Think of a singer who resonates speedily through a concert hall. This exercise illuminates how quickly sound waves can travel through solid structures, much like how vibrations move through the steel framework of tall structures.
14.17 A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m s–1).
In this exercise, we identify how pipe length and endings influence the sound produced. For a pipe closed at one end, resonance occurs only at specific odd harmonics. We can calculate the harmonics' frequencies and ascertain the excited mode from the given source frequency. This exercise details how different conditions can affect resonance within pipes, especially contrasting closed vs. open ends.
Picture a rain barrel: if you cover one side, water splashes differently than if it's entirely open. Just like sound waves react uniquely depending on the pipe's structure, this exercise pulls together how waves resonate in diverse settings for clear sound production.
14.18 Two sitar strings A and B playing the note ' Ga' are slightly out of tune and produce beats of frequency 6 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?
In this exercise, we analyze how tuning changes impact beat frequency. Given that frequency is related to tension, increasing tension raises the frequency of string B. As the beat frequency indicates the difference between the two frequencies, we can set up equations based on initial and altered conditions to solve for B's original frequency. This demonstrates how musicians tune their instruments based on frequencies, leading to harmonization.
Think of two friends singing in a choir: if one sings slightly off-key, they create dissonance. By adjusting their pitch, harmony is achieved. This exercise mirrors that experience, quantifying how changes in musical tension correlate directly with frequency adjustments.
14.19 Explain why (or how): (a) in a sound wave, a displacement node is a pressure antinode and vice versa, (b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”, (c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes, (d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and (e) the shape of a pulse gets distorted during propagation in a dispersive medium.
In these explanations, students delve deeper into wave characteristics. (a) Nodes correlate with pressure antinodes based on how sound waves propagate through compression and rarefaction, revealing their intrinsic relationship. (b) Bats utilize echolocation to navigate, relying on sound waves bouncing back to build a mental image of their surroundings, showcasing waves' practical utility. (c) Basing distinctions on timbre, two instruments may produce the same fundamental frequency yet differ in quality due to overtones and harmonic content. (d) Understanding medium propagation expectations helps clarify why solids accommodate different motion types due to particle connectivity. Finally, (e) dispersive media alters wave shape, demonstrating that speed varies across frequencies, creating distortion.
Consider translating a complex idea to a friend: while the same message may emerge, different contexts and feelings alter the understanding. This exercise parallels the interplay of wave properties, exploring how physical characteristics lead to multifaceted sound interpretations.
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
Wave Speed: The rate at which a wave travels through a medium.
Transverse Waves: Waves whose oscillations are perpendicular to the direction of wave travel.
Longitudinal Waves: Waves whose oscillations are parallel to the direction of wave travel.
Superposition Principle: The principle that states the resultant displacement of waves overlapping in a medium is the sum of their individual displacements.
See how the concepts apply in real-world scenarios to understand their practical implications.
A wave on a string demonstrating transverse wave properties.
Sound waves traveling through air showcasing longitudinal wave characteristics.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
Waves travel fast, they move with grace, v = fλ is the winning race.
Imagine two friends, Wave and Sound, racing through different mediums. Wave, who's light and quick, travels faster in water while Sound takes his time in air, reminding us that the medium matters in race.
Remember 'T for Transverse, Up and Down and L for Longitudinal, Line is Bound.'
Review key concepts with flashcards.
Term
Wave Speed Equation
Definition
Transverse Wave
Review the Definitions for terms.
Term: Wave Speed
Definition:
The speed at which a wave travels through a medium.
Term: Transverse Wave
A wave where the medium's particles move perpendicular to the direction of the wave.
Term: Longitudinal Wave
A wave where the medium's particles move parallel to the direction of the wave.
Flash Cards
Glossary of Terms