Question

If the form of a sound wave traveling through air is s(x,t)=(7.0 nm)cos(kx+(3025rad/s)t+ϕ), how much time does any given air molecule along the path take to move between displacements s=+2.0 nm and s=−2.0 nm ? Number Units

Answer 1

The time it takes for any given air molecule along the path to move between displacements s = +2.0 nm and s = -2.0 nm is approximately 0.523 ms.

Identify the relevant information:

Amplitude (A): 7.0 nm

Angular frequency (ω): 3004 rad/s

Displacements of interest: s = +2.0 nm and s = -2.0 nm

Determine the angular displacement:

The displacements of interest correspond to angles where cos(kx + ωt + φ) = ±2/7.

Using the inverse cosine function, we find these angles to be approximately ±1.23 radians.

Calculate the time for one-quarter of a period:

The time it takes to move between these displacements is one-quarter of the period (T) of the wave.

The period is related to the angular frequency by T = 2π/ω.

Therefore, T = 2π/3004 ≈ 0.00209 s.

One-quarter of the period is T/4 ≈ 0.000523 s.

Convert to milliseconds:

0.000523 s = 0.523 ms