Student Question

# A sound wave of frequency 166 Hz travels with a speed 332 ms−1 along positive x-axis in air. Each point of the medium moves up and down through 5.0 mm. Write down the equation of the wave and calculate the (i) wavelength, and (ii) wave number. How far are points which differ in phase by 45°?

The equation of the traveling wave has the following form:

y = A sin(wt ± Kx)

Where:

A, is the amplitude of the wave.

w = 2πf, is the angular frequency.

K = 2π/λ, is the constant of propagation of wave.

The sign is chosen according to the direction of propagation; positive for the negative direction and negative for the positive direction. In our case the direction is positive in x, then the equation is:

y = A sin(wt - Kx)

For the angular frequency, we have:

w = 2πf = 2π*166 = 332π

To find the propagation constant we use the relationship between the speed v and the wavelength λ:

v = λ/T

λ = v*T = v(2π/w) = 332(2π/332π) = 2 m

K = 2π/λ = 2π/2 = π m^-1

So, the wave equation is:

y = 5*10^-3 sin(332πt - πx) m

Two points located at a distance of a wavelength have a phase difference equal to 2π. For two points with a difference of phase 45° = π/4, we have:

x = λ/8 = 2/8 = 0.25 m