Question

If the form of a sound wave traveling through air is s(x, t) = (6.0 nm) cos(kx + (3021 rad/s)t + φ), how much time does any given air molecule along the path take to move between displacements s = +2.3 nm and s = -2.3 nm?

Answer 1

The given wave equation is

First, we have to determine the value of .

Now we will substitute at t=0, x = 0 and s=0, in the wave equation

or,

or,

So, our wave equation is

taking our reference point at x=0. Because we can adjust our reference point in case of sinusoidal waves.

Now at x=0 when the particle was at s = -2.3 nm, we can write

  

or,  

or,

or,

or,

This is the time taken by the particle to travel from s = -2.3 nm to s = 0.

Since it is a sinusoidal wave, the particle will take the same amount of time to reach from s=0 to s=2.3 nm.

Hence, the total time taken by the air molecule to move between s = -2.3 nm to s = 2.3 nm is 2 X 0.13 = 0.26 ms.