Superposition, Standing Waves, and Doppler Effect: NEET Problem-Solving

1. Advanced Problems in Standing Waves

Let's begin by reviewing the key formulas for standing waves on a string fixed at both ends, which you've seen before.

• Wavelength of the nth harmonic: ${\lambda }_n=\frac{2L}{n}$
• Frequency of the nth harmonic: $f_n=\frac{nv}{2L}$
• Wave speed on the string: $v=\sqrt{\frac{T}{\mu }}$

Here, $n$ is the harmonic number (1, 2, 3...), $L$ is the string length, $v$ is the wave speed, $T$ is the tension, and $\mu$ is the linear mass density (mass/length).

NEET problems often require you to combine these formulas. Let's analyze a comprehensive example.

Let's break down the logic from the video's first problem (starting at 24:56):

• Given: Speed ($v=30$ m/s), number of loops ($n=5$), frequency ($f_5=250$ Hz).

• Goal: Find length ($L$) and tension ($T$).

• Step 1: Find Wavelength (${\lambda }_5$). The fundamental relationship $v=f\lambda$ is the starting point.
  ${\lambda }_5=\frac{v}{f_5}=\frac{30}{250}=0.12\text{m}$

• Step 2: Find Length ($L$). Now use the standing wave condition $L=n\frac{{\lambda }_n}{2}$.
  $L=5\times \frac{0.12}{2}=0.3\text{m}$

• Step 3: Find Linear Density ($\mu$). To find tension, we first need $\mu$. The problem gives mass ($m=40$ g = 0.04 kg).
  $\mu =\frac{m}{L}=\frac{0.04\text{kg}}{0.3\text{m}}$

• Step 4: Find Tension ($T$). Rearrange the wave speed formula $v=\sqrt{T/\mu }$ to solve for T.
  $T=v^2\mu =(30{)}^2\times \frac{0.04}{0.3}=120\text{N}$

This example shows how you must fluidly move between different formulas to connect the given information to the desired unknown.

2. Mastering Doppler Effect Scenarios

The biggest challenge in Doppler effect problems is correctly applying the signs in the general formula:
$f^{\prime }=f\left(\frac{v\pm v_o}{v\mp v_s}\right)$
where $f^{\prime }$ is the apparent frequency, $f$ is the source frequency, $v$ is the speed of sound, $v_o$ is the observer's speed, and $v_s$ is the source's speed.

A reliable way to get the signs right is to think about the physical outcome:

• Numerator (Observer): If the observer moves towards the source, they encounter more wavefronts per second, so frequency increases. Use + $v_o$. If they move away, use - $v_o$.
• Denominator (Source): If the source moves towards the observer, it "compresses" the waves, increasing frequency. To make the fraction larger, the denominator must be smaller. Use - $v_s$. If it moves away, use + $v_s$.

Let's tackle a common NEET variation: the Doppler effect with an echo.

Problem:
A police car with a siren of frequency 1000 Hz travels towards a tall building at a speed of 20 m/s. The speed of sound in air is 340 m/s. What is the frequency of the echo heard by the driver?

Solution Strategy: This is a two-step problem.

1. Sound traveling from car to building: The car is the source, and the building is a stationary observer.
2. Sound reflecting from building to car: The building acts as a stationary source emitting the frequency it just "heard," and the moving car is now the observer.

Step 1: Frequency reaching the building ($f_{\text{building}}$)

• Source (car) is moving towards a stationary observer (building).
• Source speed $v_s=20$ m/s. Observer speed $v_o=0$.
• Use '-' in the denominator for 'towards'.
  $f_{\text{building}}=f\left(\frac{v}{v-v_s}\right)=1000\left(\frac{340}{340-20}\right)=1000\left(\frac{340}{320}\right)=1062.5\text{Hz}$

Step 2: Frequency of the echo heard by the driver ($f_{\text{echo}}$)

• The building now acts as a stationary source ($v_s=0$) emitting at $f_{\text{building}}=1062.5$ Hz.
• The driver (observer) is moving towards this stationary source at $v_o=20$ m/s.
• Use '+' in the numerator for 'towards'.
  $f_{\text{echo}}=f_{\text{building}}\left(\frac{v+v_o}{v}\right)=1062.5\left(\frac{340+20}{340}\right)=1062.5\left(\frac{360}{340}\right)\approx 1125\text{Hz}$

The driver hears an echo frequency of approximately 1125 Hz.

3. NEET Strategy: Common Mistakes and Exam Tips

To succeed under exam pressure, it's crucial to know the common traps. An excellent resource for this is a forum post that compiles tips specifically for NEET aspirants.

Here are the most critical takeaways from that resource for the topics we've covered:

• Doppler Signs: Don't just memorize the formula. Reason it out: does the frequency need to increase or decrease? That will tell you whether to make the fraction bigger or smaller.
• Standing Waves in Pipes: This is a very common point of error.
  • String / Open Pipe: All harmonics are present ($f,2f,3f,...$).
  • Closed Pipe: Only odd harmonics are present ($f,3f,5f,...$).
• Beats: The beat frequency is the difference ($|f_1-f_2|$), not the sum or average.

4. Final Practice Set

Now, apply what we've reviewed. Attempt these NEET-style questions from the provided resources.

Question 1 (Standing Waves in Pipes)
Source: LINK, Exercise 1, Q17

A closed pipe and an open pipe have their first overtones identical in frequency. Their lengths are in the ratio:
(a) 1 : 2
(b) 2 : 3
(c) 3 : 4
(d) 4 : 5

Question 2 (Doppler Effect)
Source: LINK, Section 6, Q3

A train moving at 30 m/s blows a whistle of frequency 600 Hz. The frequency heard by a stationary observer as the train approaches him is (speed of sound = 330 m/s):
(a) 660 Hz
(b) 600 Hz
(c) 550 Hz
(d) 720 Hz

Question 3 (Beats and Standing Waves)
Source: LINK, Exercise 2, Q66

A tuning fork of frequency 512 Hz makes 4 beats per second with the vibrating string of a piano. The beat frequency decreases to 2 beats per sec when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was:
(a) 510 Hz
(b) 514 Hz
(c) 516 Hz
(d) 508 Hz

Conclusion

This session was a "sprint" through the types of problems you'll encounter on the NEET from this module. We've seen how to combine formulas for standing waves, navigate the tricky signs of the Doppler effect, and use logical deduction to solve problems involving beats.

Key Takeaways:

• NEET problems often require multiple steps and the combination of several formulas. Practice breaking them down into smaller, manageable parts.
• For Doppler effect, always determine if the relative motion should increase or decrease the frequency first, then choose your signs accordingly.
• Master the conditions for harmonics and overtones in different systems (strings, open pipes, closed pipes) as this is a frequent source of error.
• When a problem gives two scenarios (like in beat frequency problems), use the first to establish possibilities and the second to eliminate the incorrect ones.

Preview of the Next Lesson:
With this, we conclude our module on waves and sound. We will now shift gears significantly. Our next module, Reflection from Plane and Spherical Mirrors, marks the beginning of our journey into Ray Optics. We will start by applying the laws of reflection to trace light rays and construct ray diagrams to locate images, a foundational skill for all of optics.