Question

The longitudinal displacement of a mass element in a medium as a sound wave passes through it is given by s = sm cos (kx – ωt). Consider a sound wave of frequency 384 Hz and wavelength 0.55 m. If sm = 14 µm, what is the displacement of an element of air located at x = 2.4 m at time t = 6.7 ms?

Answer 1

The displacement of an element of air located at x = 2.4 m at time t = 6.7 ms is 9.89 µm.

Step-by-step explanation:

The displacement of an element of air located at x = 2.4 m at time t = 6.7 ms can be calculated using the given wave function s = sm cos (kx - ωt).

First, we need to find the values for k and ω using the given information. The wave number k can be obtained from the formula k = 2π/λ. Thus, k = 2π/0.55m = 11.42 m^(-1). The angular frequency ω can be calculated using the formula ω = 2πf, where f is the frequency. Therefore, ω = 2π * 384 Hz = 2413.44 Hz.

Now, we can substitute these values into the wave function to find the displacement of the element of air at x = 2.4 m and t = 6.7 ms:

s = sm cos((11.42 m^(-1) * 2.4 m) - (2413.44 Hz * 6.7 ms))

s = 14 µm * cos(27.41 - 16.14) = 14 µm * cos(11.27) = 9.89 µm.

Answer 2

The longitudinal displacement is 1.373x10⁻⁵m

Step-by-step explanation:

Given:

f = 384 Hz

λ = 0.55 m

x = 2.4 m

t = 6.7 ms = 6.7x10⁻³s

sm = 14 µm = 14x10⁻⁶m

According the question, the displacement is:

`s=s_(m) cos(kx-wt)`

But:

`k=(2\pi )/(\lambda ) =(2\pi )/(0.55) =11.42`

`w=2\pi f=2\pi *384=2412.74`

Replacing:

`s=14x10^(-6) cos(11.42x-2412.74t)\\s=14x10^(-6) cos((11.42*2.4)-(2412.74*6.7x10^(-3)))\\ s=1.373x10^(-5) m`