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Understanding Initial Conditions
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Today, we are discussing how to apply initial conditions to D'Alembert’s solution of the wave equation. Can anyone remind me what these initial conditions represent?
They represent the initial state of the wave at time zero, right? Like its position and velocity?
Exactly! We typically have two initial conditions: the initial displacement, \( u(x, 0) = \phi(x) \), and the initial velocity, \( \frac{\partial u}{\partial t}(x, 0) = \psi(x) \). These functions help define how the wave behaves as it propagates.
So, if we set these up correctly, we can figure out what the wave looks like at any time t?
Correct! This sets the stage for deriving our solution to the wave equation.
Can anyone summarize why we differentiate the initial conditions?
Differentiating helps us establish the relationship between position and motion of the wave!
Excellent! Let's move on to how we can formulate this into a full solution of the wave equation.
Deriving Functions from Initial Conditions
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Now that we understand the initial conditions, let's learn how to derive the functions \( f(x) \) and \( g(x) \). After substituting \( t = 0 \) in D’Alembert’s solution, what do we derive?
We get \( u(x, 0) = f(x) + g(x) = \phi(x) \)!
Right! And we also have the velocity condition, which after applying provides another equation involving the derivatives of \( f \) and \( g \). Can someone express that?
It becomes \( c[f'(x) - g'(x)] = \psi(x) \)!
Perfect! This tells us how the shapes of our functions relate to the initial velocity of the wave.
So we can find \( f'(x) \) by rearranging, right?
Absolutely! This fundamental manipulation is key to solving our wave equation.
Finalizing D'Alembert's Solution
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So after deriving functions \( f \) and \( g \), what does our final formula for the wave solution look like?
It is \( u(x, t) = \frac{1}{2}{[\phi(x + ct) + \phi(x - ct)]} + \frac{1}{2c}\int \psi(s) ds \)!
Excellent! What do the two terms in this equation represent?
The first term is the wave's displacement propagating to the left and right, and the second term accounts for initial velocity!
Exactly! Understanding this helps us interpret the wave behavior in physical systems effectively.
So if we know the initial conditions, we can use this complete formula to describe the wave at any time!
That's correct! Remember, applying initial conditions is crucial for real-world applications of wave equations in physics.
Example Problem Solving
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Let's apply what we’ve learned to an example problem. Consider, we have the wave equation with initial conditions: \( u(x, 0) = \sin x \) and \( \frac{\partial u}{\partial t}(x, 0) = 0 \). How can we start?
First, we identify \( \phi(x) = \sin x \) and \( \psi(x) = 0 \).
Correct! Next, how do we set up our D'Alembert's solution with these initial conditions?
We would substitute into the formula. So it becomes \( u(x, t) = \frac{1}{2}[\sin(x + 2t) + \sin(x - 2t)] \) since the wave speed \( c = 2 \).
Exactly! And what's the final simplified form of the wave function?
After applying the identity for sine, it is \( u(x, t) = \sin x \cos(2t) \).
Fantastic! This shows how initial conditions directly impact the wave's displacement over time.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section outlines the process of applying specific initial conditions to the D’Alembert’s solution of the one-dimensional wave equation. It details how to derive functions that represent wave displacement and velocity at a particular time, providing clear mathematical representations and physical interpretations.
Detailed
Applying Initial Conditions
In this section, we explore the practical application of initial conditions to the D’Alembert’s solution of the one-dimensional wave equation, represented as:
$$
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}
$$
By defining initial conditions for displacement and velocity, we can derive the specific forms of functions under D’Alembert's framework. The initial conditions can be expressed as:
- Initial Displacement: $$ u(x, 0) = \phi(x) $$
- Initial Velocity: $$ \frac{\partial u}{\partial t}(x, 0) = \psi(x) $$
From these conditions, we can utilize the general solution provided by D’Alembert:
$$ u(x, t) = f(x + ct) + g(x - ct) $$
By substituting for time t = 0, we derive two key equations that help establish the unique forms of functions f(x) and g(x). Subsequently, this leads to the final solution form:
$$ u(x, t) = \frac{1}{2}[\phi(x + ct) + \phi(x - ct)] + \frac{1}{2c}\int \psi(s) ds $$
This comprehensive application clarifies how initial conditions influence wave behavior, demonstrating the critical link between mathematical formulation and physical interpretation.
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Initial Conditions Overview
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Suppose initial conditions are:
∂𝑢
𝑢(𝑥,0) = 𝜙(𝑥), (𝑥,0) = 𝜓(𝑥)
∂𝑡
Detailed Explanation
In this chunk, we set the initial conditions for our wave equation. The function 𝑢(𝑥,0) represents the initial displacement of the wave at any position 𝑥 when time 𝑡 is 0. It is defined by another function 𝜙(𝑥). Additionally, 𝜓(𝑥) denotes the initial velocity of the wave at time zero, which is represented by the derivative of 𝑢 with respect to time, ∂𝑢/∂𝑡.
Examples & Analogies
Think of a guitar string that is plucked. The initial displacement of the string (how far it is from its rest position) is similar to 𝜙(𝑥), while the speed at which the string moves immediately after being plucked is analogous to 𝜓(𝑥). At the moment of plucking, you can observe both the position where the string is displaced and how fast it is moving.
D'Alembert's Solution Application
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Then from D'Alembert’s solution:
At 𝑡 = 0:
• 𝑢(𝑥,0) = 𝑓(𝑥)+𝑔(𝑥) = 𝜙(𝑥) → (1)
• ∂𝑢 (𝑥,0) = 𝑐[𝑓′(𝑥)−𝑔′(𝑥)]= 𝜓(𝑥) → (2)
Detailed Explanation
Here, we apply the initial conditions directly within the framework of D'Alembert's solution for the wave equation. At time 𝑡 = 0, we know that u's value can be decomposed into two arbitrary functions, 𝑓(𝑥) and 𝑔(𝑥). The first point (1) shows the relation between these functions and the initial displacement 𝜙(𝑥). The second point (2) uses the velocity condition to relate the derivatives of these functions to the initial velocity 𝜓(𝑥). This helps to establish a system of equations that we can solve.
Examples & Analogies
Imagine the initial: when you pull a rubber band and let it go—the shape of the band when you release it is the 𝜙(𝑥), and the speed at which the band snaps back to its original shape is like 𝜓(𝑥). The functions 𝑓 and 𝑔 are the underlying functions that describe how the rubber band behaves after release, representing the way it’s affected by initial conditions.
Finding Functions from Initial Conditions
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From (1):
𝑓(𝑥)+ 𝑔(𝑥) = 𝜙(𝑥) ⇒ 𝑔(𝑥) = 𝜙(𝑥)−𝑓(𝑥)
Differentiate and substitute into (2):
𝑐[𝑓′(𝑥)− (𝜙′(𝑥)−𝑓′(𝑥))] = 𝜓(𝑥) ⇒ 2𝑐𝑓′(𝑥)= 𝜓(𝑥)+ 𝑐𝜙′(𝑥)⇒ 𝑓′(𝑥)
1 1
= 𝜓(𝑥)+ 𝜙′(𝑥)
2𝑐 2
Detailed Explanation
In this section, we derive function 𝑔(𝑥) using the relationship established in (1) by isolating it. We then differentiate the equation for the velocity condition before substituting this result into the second equation from the initial conditions. This leads to a new equation for the derivative of 𝑓(𝑥), which can then be further manipulated to isolate 𝑓′(𝑥). This step is crucial as it allows us to express 𝑓′(𝑥) in terms of known functions, which can then be integrated to find 𝑓 and subsequently 𝑔.
Examples & Analogies
Returning to our rubber band analogy, discovering 𝑓 and 𝑔 is like determining the exact shape of the rubber band once it's released based on its starting point and velocity. If we know how far we pulled it back and how fast it snaps back, we can calculate every position it takes on its way back to its original position.
Integration to Find Functions
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Integrate to get 𝑓(𝑥), then use 𝑔(𝑥) = 𝜙(𝑥)−𝑓(𝑥) to find 𝑔(𝑥).
Detailed Explanation
Now, we take the expression we found for 𝑓′(𝑥) and integrate it to find 𝑓(𝑥). This gives us the exact functional form of the displacement related to 𝑓. Once we have that, it is straightforward to find 𝑔(𝑥) by substitutively using the relation we derived earlier, where 𝑔(𝑥) is defined as the difference between 𝜙(𝑥) and 𝑓(𝑥). This step is the finalization of the component functions required for the complete D'Alembert solution.
Examples & Analogies
Continuing with the rubber band metaphor, once you calculate how the rubber band moves back into shape (represented by 𝑓(𝑥)), you can also easily determine how much of that initial displacement remains and flows back into the overall wave motion (represented by 𝑔(𝑥)). Essentially, it’s about combining what we know about the initial conditions to predict the entire movement of the wave thereafter.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Initial Conditions: Define the state of the wave at time t=0, essential for solving the wave equation.
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D'Alembert's Solution: The formula that provides the general solution for waves based on initial displacement and velocity.
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Wave Functions: Functions \( f \) and \( g \) that represent the right and left-moving components of the wave.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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An example applying initial conditions with \( u(x, 0) = \sin x \) and \( \frac{\partial u}{\partial t}(x, 0) = 0 \) resulting in \( u(x, t) = \sin x \cos(2t) \).
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Solving for wave displacement when initial conditions are defined helps to visualize how waves behave over time.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To find the waves at time zero, learn \( \phi \) and \( \psi \) without fear.
📖 Fascinating Stories
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Imagine a calm lake. Before a stone is thrown, the water's surface is perfectly still (initial displacement). As the stone strikes, ripples spread out – these represent the wave's initial motion (initial velocity). This illustrates how waves evolve from their starting conditions.
🧠 Other Memory Gems
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Remember D'Alembert as D-A-M (Displacement, Acceleration, Motion) when applying initial conditions.
🎯 Super Acronyms
D.A.W (D'Alembert's Applied Waves) – the essentials of applying initial conditions to wave equations.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Initial Conditions
Definition:
Specific values assigned to the wave at time t=0, necessary for solving the wave equation.
-
Term: D'Alembert’s Solution
Definition:
A formula providing the solution to the one-dimensional wave equation involving arbitrary functions of displacement and velocity.
-
Term: Displacement
Definition:
The distance and direction a wave is offset from its rest position.
-
Term: Velocity
Definition:
The rate of change of displacement with respect to time, indicating how fast the wave propagates.
-
Term: Wave Propagation
Definition:
The movement of wave energy through space and time.