Animated Solution for Physics - Waves: A source of sound of frequency $600\text{ Hz}$ is placed inside water. The speed of sound in water is $1500\text{ m/s}$ and in air it is $300\text{ m/s}$. The frequency of sound recorded by an observer who is standing in air is

Imagine you are standing on the edge of a quiet swimming pool. Deep below the surface, an underwater speaker is playing a pure tone of frequency $600\text{ Hz}$. As the sound waves travel upwards, they cross the boundary between water and air, eventually reaching your ears.

We are given the following physical parameters: - Frequency of the source in water: $f=600\text{ Hz}$ - Speed of sound in water: $v_{\text{water}}=1500\text{ m/s}$ - Speed of sound in air: $v_{\text{air}}=300\text{ m/s}$

Our goal is to determine the frequency of the sound wave recorded by you, the observer standing in the air.

Source vs. Medium

To solve this problem, we must understand a fundamental truth of wave mechanics: frequency is a characteristic of the source, not the medium.

When a wave is generated, the source vibrates at a specific rate. For our speaker, it pushes the surrounding water molecules back and forth exactly $600$ times every second. These water molecules then push the adjacent molecules, propagating the wave outward.

When the wave reaches the water-air interface, the water molecules at the boundary vibrate at $600\text{ Hz}$ and force the adjacent air molecules to vibrate at the exact same rate. The air molecules have no choice but to oscillate at the frequency of the driving force.

Therefore, as a wave transitions from one medium to another, its frequency remains absolutely constant:

While the frequency remains unchanged, the speed of the wave is determined strictly by the properties of the medium (such as elasticity and density). Since sound travels much slower in air ($300\text{ m/s}$) than in water ($1500\text{ m/s}$), the wave must slow down.

To maintain the fundamental wave relationship:

the wavelength $\lambda$ must adjust proportionally to the change in speed.

Let's calculate the wavelength in both media to see this beautiful symmetry.

In water:

In air:

Notice that as the wave enters the air, its speed decreases by a factor of $5$ (from $1500\text{ m/s}$ to $300\text{ m/s}$), and consequently, its wavelength also shrinks by a factor of $5$ (from $2.5\text{ m}$ to $0.5\text{ m}$).

This perfect scaling ensures that the frequency—the number of wave crests passing a point per second—remains exactly $600\text{ Hz}$.

Since the frequency of the sound wave does not change when crossing the boundary, the observer in the air hears the exact same frequency as emitted by the source.

Thus, the correct option is (d).