A whistle giving out 450 Hz approaches a stationary observer at a speed of 33 m/s. The frequency hea

Q: Why do we subtract the source velocity in the denominator instead of adding it?

When the source moves towards the observer, it 'chases' its own wavefronts, compressing them. This compression reduces the wavelength, which increases the frequency. Mathematically, subtracting $v_s$ from $v$ in the denominator makes the denominator smaller, which increases the overall value of the fraction, resulting in a higher apparent frequency ($f' > f$).

Q: Does the speed of sound in air change because the whistle is moving?

No, the speed of sound in air is a property of the medium itself (determined by temperature, pressure, and molecular mass of air). It is completely independent of the motion of the source or the observer.

Q: What would happen if the observer was also moving towards the source?

If the observer was also moving towards the source with some velocity $v_o$, the apparent frequency would increase even further. The formula would become $f' = f \left( \frac{v + v_o}{v - v_s} \right)$.

Q: What if the source speed $v_s$ is greater than the speed of sound $v$?

If $v_s > v$, the source is traveling faster than sound (supersonic). In this case, the denominator becomes negative, and the standard Doppler formula breaks down. Instead, a shock wave (sonic boom) is created in the shape of a cone.

Q: Is the frequency of the source itself changing?

No, the whistle continues to vibrate and emit sound at its natural frequency of $450\text{ Hz}$. The change in frequency is purely relative and only exists from the perspective of the observer.

Question

A whistle giving out $450	ext{ Hz}$ approaches a stationary observer at a speed of $33	ext{ m/s}$. The frequency heard by the observer (in $	ext{Hz}$) is (Speed of sound $= 330	ext{ m/s}$)

Options:

(A) 409

(B) 429

(C) 517

(D) 500

Sigma Percentile · Accepted Answer

### The Magic of Sound and Motion

Have you ever stood by the side of a road and listened to an ambulance or a police car speed past you?

If you have, you've undoubtedly noticed a fascinating phenomenon: as the vehicle rushes towards you, the siren sounds high-pitched and piercing.

But the very instant it passes you and begins to move away, the pitch drops dramatically to a lower, flatter tone.

This is not a trick of your mind, nor is the driver changing the siren's settings.

This is the **Doppler Effect**, one of the most fundamental and beautiful wave phenomena in physics.

Today, we are going to explore this concept through a classic JEE problem involving a moving whistle and a stationary observer.

---

### Analyzing the Setup

Let's break down the physical variables given to us in the problem:

* The original frequency of the whistle, which we will call $f$, is **$450	ext{ Hz}$**.
* The speed of the source (the whistle), $v_s$, is **$33	ext{ m/s}$**.
* The observer is completely stationary, meaning their velocity $v_o = 0$.
* The speed of sound in air, $v$, is **$330	ext{ m/s}$**.

Because the source is moving directly towards the stationary observer, the sound waves emitted in the forward direction are compressed.

This compression decreases the wavelength of the sound waves reaching the observer's ear.

Since the speed of sound in the air remains constant, a shorter wavelength directly translates to a higher frequency.

---

### The Master Equation

The general formula for the Doppler effect in sound is:

$$f' = f \left( \frac{v \pm v_o}{v \mp v_s} ight)$$

Where:
* $f'$ is the apparent frequency heard by the observer.
* $f$ is the source frequency.
* $v$ is the speed of sound in the medium.
* $v_o$ is the speed of the observer.
* $v_s$ is the speed of the source.

Since our observer is stationary ($v_o = 0$), the numerator simplifies to just $v$.

And because the source is approaching the observer, we expect the frequency to increase.

To make the fraction larger, we must make the denominator smaller.

Therefore, we choose the minus sign in the denominator:

$$f' = f \left( \frac{v}{v - v_s} ight)$$

---

### Step-by-Step Calculation

Let's substitute our values into this elegant formula:

$$f' = 450 \left( \frac{330}{330 - 33} ight)$$

First, let's simplify the denominator:

$$330 - 33 = 297$$

Now, our equation looks like this:

$$f' = 450 \left( \frac{330}{297} ight)$$

At first glance, dividing $330$ by $297$ might seem tedious.

But look closely! Both numbers are multiples of $11$, and even more conveniently, multiples of $33$:

$$\frac{330}{33} = 10$$
$$\frac{297}{33} = 9$$

So, the fraction simplifies beautifully to:

$$\frac{10}{9}$$

Let's plug this back into our main equation:

$$f' = 450 	imes \frac{10}{9}$$

Since $450$ is perfectly divisible by $9$:

$$\frac{450}{9} = 50$$

We are left with a simple multiplication:

$$f' = 50 	imes 10 = 500	ext{ Hz}$$

And there we have it! The apparent frequency heard by the observer is exactly **$500	ext{ Hz}$**.

---

### Physical Intuition & Sanity Check

Always perform a quick sanity check when solving wave mechanics problems.

Our calculated frequency ($500	ext{ Hz}$) is higher than the original frequency ($450	ext{ Hz}$).

This perfectly aligns with our physical intuition: an approaching source must produce a higher pitch.

If you had accidentally added $v_s$ in the denominator instead of subtracting it, you would have obtained a frequency lower than $450	ext{ Hz}$, signaling an immediate error.

Mastering these small checks is what separates top scorers from the rest!

Animated Solution for Physics - Waves: A whistle giving out $450 Hz$ approaches a stationary observer at a speed of $33 m/s$ . The frequency heard by the observer (in $Hz$ ) is (Speed of sound $= 330 m/s$ )

Select Answer:

Visualized Solution

Visualizing the Physical Setup

The Doppler Effect Formula

Substituting the Given Values

Calculating the Denominator

Simplifying the Fraction

Finding the Apparent Frequency

Analyzing the Result

The Sigma Insight: Doppler Effect

The Magic of Sound and Motion

Analyzing the Setup

The Master Equation

Step-by-Step Calculation

Physical Intuition & Sanity Check

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