Animated Solution for Physics - Waves: A train S1, moving with a uniform velocity of 108 km/h, approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2, as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is ______. [Speed of the sound = 330 m/s]

Imagine standing near a railway track. You hear the whistle of a train approaching you—the pitch sounds higher than it actually is. This is the classic Doppler Effect. But what happens when there are two trains, and you are also moving? The sound waves from both trains reach you simultaneously, but because of the different relative velocities, they arrive with slightly different frequencies. When these two distinct frequencies superimpose, they create a rhythmic pulsing sound known as beats.

In this fascinating problem, we are tasked with finding exactly how many beats an observer hears per second. We have two trains, $S_1$ and $S_2$, both blowing whistles at a natural frequency of $120\text{ Hz}$. The observer $O$ is moving towards a stationary train $S_2$, while another train $S_1$ is moving towards $S_2$ from a perpendicular direction.

Before we dive into the heavy physics, let's clean up our units. In physics, consistency is everything. The velocities are given in $\text{km/h}$, but the speed of sound is in $\text{m/s}$. Let's convert them:

Train $S_2$ is standing on the platform, so its velocity is $0\text{ m/s}$. Now, we are ready to decode the sound waves.

The Stationary Source

Let's start with the easier part of the problem: the sound coming from train $S_2$.

Train $S_2$ is stationary, and the observer $O$ is moving directly towards it at $10\text{ m/s}$. Because the motion is purely one-dimensional along the line connecting them, we can apply the standard Doppler formula directly.

When an observer moves towards a stationary source, they intercept more sound wavefronts per second than if they were standing still. This increases the apparent frequency. The formula for this scenario is:

Here, $v$ is the speed of sound ($330\text{ m/s}$), $v_O$ is the velocity of the observer, and $f_0$ is the original frequency. Let's plug in our values:

Calculating this gives us:

This is the first note our observer hears. It's slightly higher than the original $120\text{ Hz}$, which makes perfect physical sense because the observer is rushing towards the source.

Finding the Line of Sight

Now comes the tricky part: the sound from train $S_1$.

Train $S_1$ is moving horizontally, and the observer $O$ is moving vertically. They are not moving directly towards each other. This is a classic case of the 2D Doppler Effect.

The most critical rule of the Doppler effect is this: Only the relative motion along the line joining the source and the observer affects the frequency. Any motion perpendicular to this line (transverse motion) does not compress or stretch the sound waves in classical acoustics.

Therefore, our first job is to establish this "line of sight" between $O$ and $S_1$.

Looking at the geometry, $S_2$ is at the right angle of a triangle. The distance from $O$ to $S_2$ is $600\text{ m}$, and the distance from $S_2$ to $S_1$ is $800\text{ m}$. This forms a classic 3-4-5 right-angled triangle!

Using the Pythagorean theorem, the distance between the observer $O$ and train $S_1$ is:

The Crucial Step

Now that we have our line of sight, we need to find out how fast $O$ and $S_1$ are moving along this specific line. To do this, we need the angles.

Let $\theta$ be the angle at the observer's position between their direction of motion (towards $S_2$) and the line of sight to $S_1$.

This corresponds to an angle of $53^{\circ }$.

Similarly, let $\varphi$ be the angle at $S_1$'s position between its direction of motion (towards $S_2$) and the line of sight to $O$.

This corresponds to an angle of $37^{\circ }$.

Now, we resolve the velocities. The component of the observer's velocity directed towards $S_1$ is:

The component of train $S_1$'s velocity directed towards the observer is:

We finally have the effective velocities along the line of sight. The observer is moving towards the source at $6\text{ m/s}$, and the source is moving towards the observer at $24\text{ m/s}$.

Since both are approaching each other, the apparent frequency will increase significantly. The general Doppler formula when both are in motion is:

Notice the signs: a plus in the numerator because the observer's approach increases frequency, and a minus in the denominator because the source's approach also increases frequency.

Let's substitute our resolved components:

Calculating this yields:

This is the second note.

The observer is now hearing two distinct frequencies simultaneously: $f_1=131.76\text{ Hz}$ and $f_2=123.63\text{ Hz}$. When two sound waves of slightly different frequencies interfere, they produce beats.

The beat frequency is simply the absolute difference between the two apparent frequencies:

And there we have it! The observer hears approximately 8.13 beats per second. This problem is a beautiful demonstration of how geometry and kinematics intertwine in wave mechanics. By carefully breaking down the 2D motion into 1D components along the line of sight, a complex scenario becomes an elegant application of fundamental principles.