Question

Assuming the frequency stays the same, will a sound wave have a larger or smaller wavelength when it passes from room temperature air (20°C) to water? Explain your answer with math, words, or both.​

Answer 1

Answer:

The wavelength of this sound wave would be longer in water than in the air.  

Explanation:

Let $f$ denote the frequency of this sound wave (standard unit: $\rm s^{-1}$.)

If the speed of sound in a particular medium is $v$ (standard unit: ${\rm m \cdot s^{-1}}$,) the wavelength $\lambda$ of this wave in that medium would be:

$\begin{aligned} \lambda = \frac{v}{f} && \genfrac{}{}{0}{}{(\text{standard unit: {\rm m \cdot s^{-1}} })}{(\text{standard unit: {\rm s^{-1}} })}\end{aligned}$.

Let $v_\text{water}$ denote the speed of sound in water and let $v_\text{air}$ denote the speed of sound in the air at room temperature.

The wavelength of this sound wave in water would be:

$\displaystyle \lambda_{\text{water}} = \frac{v_{\text{water}}}{f}$.

The wavelength of this sound wave in the air at room temperature would be:

$\displaystyle \lambda_{\text{air}} = \frac{v_{\text{air}}}{f}$.

Fact: the speed of sound in water (a liquid) is greater than the speed of sound in air at room temperature. In other words:

$v_{\text{water}} > v_{\text{air}}$.

Given that $f > 0$:

$\begin{aligned} \frac{v_{\text{water}}}{f} > \frac{v_{\text{air}}}{f}\end{aligned}$.

Therefore:

$\begin{aligned} \lambda_{\text{water}} > \lambda_{\text{air}}\end{aligned}$.