14.4. THE SPEED OF A TRAVELLING WAVE

To determine the speed of propagation of a travelling wave, we can fix our attention on any particular point on the wave (characterised by some value of the phase) and see how that point moves in time. It is convenient to look at the motion of the crest of the wave. Fig. 14.8 gives the shape of the wave at two instants of time, which differ by a small time interval ∆t. The entire wave pattern is seen to shift to the right (positive direction of x-axis) by a distance ∆x. In particular, the crest shown by a dot (•) moves a distance ∆x in time ∆t. The speed of the wave is then ∆x/∆t.

When we want to understand how fast a wave travels, we focus on a specific point on the wave, often the crest. By observing the wave at two different times, we see how far that crest has moved in a very short duration (∆t). The distance the crest moves during that time (∆x) gives us the speed of the wave when we divide the distance by the time.

Think of a floating leaf on a wave in a pond. If you watched the leaf for a short time, you’d see it rise and fall as the wave passes, but if you measured how far it moved horizontally across the surface, you’d get a sense of the wave's speed.

The motion of a fixed phase point on the wave is given by kx – ωt = constant (14.10). Thus, as time t changes, the position x of the fixed phase point must change so that the phase remains constant. Thus, kx – ωt = k(x+∆x) – ω(t+∆t) or k∆x – ω∆t = 0 Taking ∆x, ∆t vanishingly small, this gives ω = v k.

To understand wave motion mathematically, we can use the equation kx – ωt = constant. Here, k represents the wave number, and ω is the angular frequency. This equation implies that as time changes, the position of a wave point also changes to keep the wave pattern consistent. By taking the limits as the shifts become infinitesimally small, we find a relationship where the speed of the wave (v) is directly related to the angular frequency (ω) and wave number (k).

Imagine a dancer performing twirls on stage. The dancer's position changes over time, but the flow of their movements remains smooth. In this analogy, the dancer's position is like the wave's position, and the rhythm of their steps aligns with the wave's frequency and wavelength.

Relating ω to T and k to λ, we get v = λ/T = fλ (14.12). Eq. (14.12), a general relation for all progressive waves, shows that in the time required for one full oscillation by any constituent of the medium, the wave pattern travels a distance equal to the wavelength of the wave.

Here we connect the speed of a wave to its wavelength (λ) and the frequency (f). The equation v = λ/T shows that the speed of a wave depends on both how long the wave is (wavelength) and how frequently it oscillates (frequency). Essentially, in the time it takes for one oscillation, the wave travels a distance equal to its wavelength.

Consider a train moving along a track. If you measure the distance between each train car (the wavelength) and how fast the train runs (the speed), you can also think about how often the train passes noted points (the frequency). Together, these measures help you understand the overall motion of the train.

The speed of a mechanical wave is determined by the restoring force setup in the medium when it is disturbed and the inertial properties (mass density) of the medium. The speed is expected to be directly related to the former and inversely to the latter. For waves on a string, the restoring force is provided by the tension T in the string. The inertial property will in this case be linear mass density µ, which is mass m of the string divided by its length L.

To find the speed of waves on a string, we consider two main factors: the tension in the string and the mass per unit length (linear mass density). The greater the tension, the faster the wave travels because the string can return to its resting position more quickly. Conversely, if the string is heavier (higher mass density), the speed decreases because more energy is needed to move the string.

Think of a tightrope walker. A tight and well-strung wire allows for a fast response to movements, similar to how higher tension in a string allows faster wave propagation. If the wire were heavier and thicker, it would sway more slowly, representing how mass density affects wave speed.

For longitudinal waves, the speed is determined by the bulk modulus of the medium (B) and its density (ρ). The general formula for longitudinal waves is given by v = B/ρ (14.19). This shows that the speed of sound waves in gases, liquids, and solids is different due to their unique elastic and inertial properties.

In a longitudinal wave, such as sound traveling through air, the speed depends on how compressible the medium is (bulk modulus) and how heavy the medium is (density). The formula v = B/ρ highlights this relationship. The stiffer the medium and the less it weighs, generally, the faster the wave travels.

Imagine blowing through a straw into a glass of water. The water (liquid) reacts quickly to your breath, showing sound's fast travel in liquids due to better compressibility. Now, if you tried this with a dense material like steel, sound travels even faster due to its rigidity and lower compressibility compared with gases.

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