The figure shows the output from a pressure monitor mounted

The figure shows the output from a pressure monitor mounted at a point along the path taken by a sound wave of a single frequency travelling at 346 m/s through air with a uniform density of 1.41 kg/m^3. The vertical axis scale is set by delta ps = 4.20 mPa. If the displacement function of the wave is written as s(x, t) = s_m cos(kx - omega t), what are (a) s_m, (b) k, and (c) omega? The air is then cooled so that its density is 1.55 kg/m^3 and the speed of a sound wave through it is 320 m/s. The sound source again emits the sound wave at the same frequency and same pressure amplitude. What now are (d) s_m, (e) k, and (f) omega?

Solution

A)

The displacement amplitude is calculated as follows:

      Sm = dP*T/v*p*2*pi = [4.2x10^-3][2.4*10^-3] / [343][1.41][2][pi] = 3.32e-9 m

b)

the wavenumber is,

   k = 2*pi/T*v = 2*pi/2.4*10^-3*343 = 7.63 rad/m

c)

angular frequency: w = 2*pi/T = 2*pi/2.4x10^-3 = 2616.67 rad/s = 2.62e+3 rad/s

d)

The displacement amplitude is calculated as follows:

      Sm = dP*T/v*p*2*pi = [4.2x10^-3][2.4*10^-3] / [320][1.55][2][pi] = 3.236e-9 m

(e)

the wavenumber is,

   k = 2*pi/T*v\'

(f)

angular frequency: w = 2*pi/T = 2*pi/2.4x10^-3 = 2616.67 rad/s = 2.62e+3 rad/s