# Is the amplitude of vibration of all particles equal in a sound wave?

[Assume ideal conditions and simple harmonic wave]

My book showed resemblance in a lot of equations related to both transverse and longitudinal waves although they work very differently.
the displacement of particle's equation in a transverse wave can be:

$$y=A\sin(\omega t-kx)$$ where A is the amplitude.

while the displacement equation in a sound/longitudinal wave can be taken as:

$$s=s_0\sin(\omega t-kx)$$ where $$s_o$$ is the amplitude

Looking at these, I obviously concluded that just like the amplitude of all the particles in a transverse wave is A, the amplitude of the vibrations of all the particles in a sound wave must be equal as well with the value being, $$s_0$$

But in one of the problems, I found that this is not true. Why is it so? Is my book wrong about this or is there some error in my interpretation and understanding?

For reference, here is the problem(It was a matrix match type problem, so I'm just going to mention this specific problem and not the other parts):

Amplitude of vibration of all particles are equal in which of the following types of waves?
The options given were:

1. Stationary waves
2. Plane simple harmonic transverse waves
3. Sound waves
4. Standing waves in an open organ pipe
5. Plane simple harmonic longitudinal waves

The answer given was:

2, 5 I understand how it is 2 and 5 but why is it not 3(that is, sound waves)

• Please add that question Mar 30 at 17:01
• @AdityaPrakash I have edited my post and added the problem. Mar 30 at 17:17

## 1 Answer

Indeed all the particles have the same amplitude of vibration in both transverse and longitudinal waves.

The exception being standing waves, which are a combination of waves. Standing waves produce nodes and anti-nodes, nodes being point of zero amplitude, and anti-nodes being points of maximum amplitude.

The equation of a standing wave is as follows (it is formed by adding the wave function of 2 compatible waves):

$$A=\left[A_{o}\sin\left(kx-\phi\right)\right]\sin\left(ωt-\phi\right)$$

The term $$\left[A_{o}\sin\left(kx-\phi\right)\right]$$ represents the variation of amplitude with distance.

The question asked in your book may very well be referring to standing waves

Here is a visualization of standing waves to help explain it better, the red dots are nodes, and the points of maximum amplitude are anti-nodes. The faded red and blue waves show the waves that combined to form the standing wave.

In your book, sound waves have been included in the varying amplitude case due to the fact that sound waves are radial waves, i.e. the spread spherically and uniformly from a source. Now, this type of wave has an amplitude that varies with distance.

Using the inverse square law, we see that if the initial intensity of the sound was $$I_{o}$$, then at a distance r, it will reduce by a factor of $$\frac{1}{r^{2}}$$

Now, we have an equation that relates intensity to amplitude, which says that $$I\ ∝\ A^{2}$$.

Hence, we can conclude that the amplitude varies as a function of r, and deceases as r increases. This is what i believe your book was referring to.

• Yes! that's what I thought. Maybe then this might be an error in my book. Thank you! Mar 30 at 17:18
• @PrajwalTiwari I have made some changes in my answer to include the sound waves :) Mar 30 at 17:32
• Oh yes! This completely slipped out of my mind. Thanks a lot! :) Mar 30 at 17:33