Longitudinal Wave Equation meaning and derivation

[Enterprise D]

Hi,

I have recently been studying waves, and I understand the transversal wave formula

y=Asin(w(t-x/v))

which gives the y coordinate of a point at x along the x-axis in the instant t.

However, Wikipedia (http://en.wikipedia.org/wiki/Longitudinal_wave) gives this as the equation for longitudinal waves too.

If this is true, what does y represent? The displacement of a point relative to what? Also, does anyone know how to derive this formula specifically for longitudinal waves?

Thanks very much!

 

[Blue chocolate]

For something such as a sound wave, Y represents the pressure at point x, at time t. It's a bit more challenging to visualize, but the formula is applied the same way.

 

[Philip Wood]

For a longitudinal wave, y can also represent the displacement of particles in the medium. The displacement is to and fro parallel to the direction of propagation of the waves, whereas for a transverse wave the displacement is at right angles to the direction of propagation. The equation you quote can be used to represent either sort of wave.

 

[Enterprise D]

Hi, thanks for your replies!

So am I right in saying that when the wave reaches a point at rest, the point starts to move with a harmonic motion in which its point of rest becomes the middle point of the motion (i.e. where its speed is greatest)?

How can we show that the equation describes this kind of motion?

Thanks!

 

[pumila]

For any wave that is neither damped nor forced and at constant amplitude, the energy of deflection (spring or pressure energy) plus the energy of motion (kinetic energy) is a constant. Hence when the energy of deflection is zero the kinetic energy is at a max, and vice-versa. As you state.