Waves (LAQs)

Physics-2 | 1. Waves – LAQs:
Welcome to LAQs in Chapter 1: Waves. This page features the key FAQs for Long Answer Questions. Each answer is provided in simple English, with a Telugu explanation, and formatted according to the exam style. This will support your preparation and help you secure top marks in your final exams.

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LAQ-1 : Explain the formation of stationary waves in stretched strings and hence deduce the equations for first, second and third harmonics and also deduce the laws of transverse waves is stretched strings.

Introduction

Stationary waves in stretched strings are a unique phenomenon where two identical waves travel in opposite directions and meet, forming a distinct wave pattern. These waves are essential in understanding harmonics, which are distinct patterns created by stationary waves on strings.

Formation of Stationary Waves

1. Stationary Waves Creation: Stationary waves are created when two identical waves with the same amplitude and wavelength travel in opposite directions and overlap or superimpose on each other.
2. Reflection and Overlapping: In stretched strings, waves traveling in one direction are reflected back, resulting in a stationary wave system that remains fixed in space.

Understanding Nodes and Antinodes

1. Antinodes: Points in the stationary wave where the amplitude is maximum, occurring at x = 0, λ/2, λ, 3λ/2, where the value of cos(2πx/λ) is ±1.
2. Nodes: Points where the amplitude is minimum (zero), occurring at x = λ/4, 3λ/4, 5λ/4, where the value of cos(2πx/λ) is zero.
3. Distance Between Nodes and Antinodes: The distance between any two successive antinodes or nodes is λ/2, and between any antinode and node is λ/4.

Harmonics

1. Harmonics on Vibrating Strings: In a vibrating string, the audible frequencies are called harmonics.
2. First Harmonic: Also known as the fundamental frequency or first resonance, occurs when one full wave fits on the string, given by f1​ = v/2L.
3. Second Harmonic: Occurs when two full waves fit on the string, expressed as f2​ = 2v/2L = v/L.
4. Third Harmonic: Involves three full waves fitting on the string, calculated as f3​ = 3v/2L.
5. Harmonic Formula: Here, v represents the speed of the wave on the string, and L represents the length of the string.

Summary

Stationary waves form when two identical waves traveling in opposite directions superimpose on each other, creating points of maximum amplitude (antinodes) and minimum amplitude (nodes). This phenomenon in stretched strings forms a pattern known as harmonics. The first harmonic corresponds to one full wave fitting on the string, and subsequent harmonics follow a similar pattern. The frequency of these harmonics is determined by the formula f = nv/2L, where n is the harmonic number, v is the wave speed, and L is the string length.

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LAQ-2 : Explain the formation of stationary waves in air column enclosed in open pipe. Derive the equation for the frequencies of harmonics produced.

Introduction

In the world of sound and music, an open pipe is a term used to describe a tube that is open at both ends. These open pipes are often used in musical instruments, like flutes or some kinds of organs. When sound waves are sent through an open pipe, they bounce back from the far end and mix with incoming waves to create what we call stationary waves. These stationary waves result in the formation of harmonics.

Understanding Stationary Waves and Harmonics in an Open Pipe

1. When sound waves are sent through an open pipe, they reflect back from the far end.
2. The incoming and reflected waves, which are of the same frequency but traveling in opposite directions, combine or superimpose to form stationary waves.
3. These stationary waves create what we call harmonics. The first harmonic, or fundamental frequency, is formed with antinodes at the two ends of the pipe and a node in between.

Calculating Harmonics

1. The vibrating length ‘l’ of the pipe corresponds to half the wavelength (λ/2) of the first harmonic. Therefore, the wavelength of the first harmonic is twice the length of the pipe (λ₁​ = 2l).
2. The fundamental frequency (v₁​) can be calculated using the formula v_1​ = v/λ_1​, where v is the velocity of sound in air. By substituting λ_1​ = 2l in the formula, we get v₁​ = v/2l.
3. The second harmonic (first overtone) will have an additional node and antinode than the fundamental. The wavelength of the second harmonic (λ₂​) will be the length of the pipe (λ_2​ = l). Therefore, the frequency of the second harmonic (v₂​) will be v₂ ​= v/λ₂​ = 2v₁​.
4. Similarly, the third harmonic (second overtone) will have three nodes and four antinodes. The wavelength of the third harmonic (λ₃​) will be 2l/3. Therefore, the frequency of the third harmonic (v₃​) will be v₃​ = 3v₁​.

Summary

In an open pipe, sound waves reflect and create stationary waves, forming harmonics. These harmonics are calculated based on the length of the pipe and the velocity of sound in air. The fundamental frequency or the first harmonic has the frequency of v₁ ​= v/2l. The second and third harmonics (and so on) can be found in a similar manner, with frequencies of 2v₁​, 3v₁​, etc. Thus, the ratio of harmonic frequencies in an open pipe is 1:2:3 and so on. This understanding of harmonics in open pipes forms the basis for the creation of musical notes in many wind instruments.

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LAQ-3 : How are stationary waves formed in closed pipes ? Explain the various modes of vibrations and obtain relations for their frequencies.

Introduction

Organ pipes are devices designed to create sound by utilizing the phenomenon of resonance. They come in two types: open and closed organ pipes. Open organ pipes are open at both ends, while closed organ pipes are sealed at one end. Despite their different constructions, both types serve the same purpose: to produce sound. The sound waves in these pipes form stationary waves due to the reflection and superposition of the incident sound waves.

1. Open and Closed Organ Pipes:
   • Open organ pipes, also known as open mouth organ pipes, are open at both ends.
   • Closed organ pipes, also referred to as closed mouth organ pipes, are open at one end and closed at the other.
2. Formation of Stationary Waves: Stationary waves are formed in these pipes by the reflection and superposition of the incoming sound waves.
3. Calculation of Frequencies:
   • Let’s assume that ‘L’ is the length of the organ pipe and ‘v’ is the velocity of sound.
   • The frequency of the sound wave in the first mode of vibration is calculated by the formula: $$f_1 = v / 2L$$
   • For the second mode of vibration, the frequency of the sound wave is calculated by the formula: $$f_2 = v / L$$
   • Similarly, for the third mode of vibration, the frequency of the sound wave is calculated by the formula: $$f_3 = 3v / 2L$$

Summary

Open and closed organ pipes, or mouth organ pipes, are devices designed to create sound using the principle of resonance. These pipes, despite their different constructions, are used to produce sound by forming stationary waves through the reflection and superposition of the incident sound waves. The frequencies of these sound waves in various modes of vibration are calculated using the length of the organ pipe and the velocity of sound. For example, the frequencies in the first, second, and third modes of vibration are given by v/2L, v/L, and 3v/2L, respectively. Understanding the fundamentals of these organ pipes and how they work is key to solving related problems.

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LAQ-4 : What is Doppler Effect ? Obtain an expression for the apparent frequency of sound heard when the source is in motion with respect to an observer at rest.

Introduction

The Doppler Effect is a phenomenon observed when there is a change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. This effect can be observed for all types of waves, including sound waves. It explains why the pitch of a moving siren seems higher as it approaches an observer and lower as it moves away.

Obtaining an Expression for Apparent Frequency

When the source is in motion with respect to an observer at rest, the apparent frequency (f^′) of the sound heard differs from the actual frequency (f) emitted by the source. Assume that the velocity of sound in the medium is v, the velocity of the source is vs​, and the source is moving towards the observer.

The wavelength of the sound waves decreases when the source moves towards the observer, leading to an increase in the frequency of the sound heard. The apparent frequency can be expressed as:

$$f’ = \frac{v}{v – v_s} \times f$$

Here, v is the speed of sound in the medium, vs​ is the speed of the source towards the observer, and f is the original frequency of the sound.

Key Points

1. Doppler Effect: A change in frequency or wavelength of a wave for an observer moving relative to the source of the wave.
2. Apparent Frequency: The frequency of sound as perceived by an observer, which changes when the source of the sound is moving.
3. Expression for Apparent Frequency: The formula $$f’ = \frac{v}{v – v_s} \times f$$ calculates the apparent frequency when the source is moving towards a stationary observer.

Summary

The Doppler Effect is a crucial concept in understanding how the motion of a sound source relative to an observer affects the perceived sound frequency. By applying the formula for the apparent frequency, one can calculate how the pitch of the sound changes as the source moves towards or away from the observer. This effect has practical applications in various fields, including astronomy, radar, and medical imaging.

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LAQ-5 : What is Doppler Shift ? Obtain an expression for the apparent frequency of sound heard when the observer is in motion with respect to a source at rest.

Introduction

Doppler Shift refers to the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It’s an important concept in fields like astronomy, medical imaging, and radar technology. In this article, we’ll explain the Doppler Shift in the context of sound waves, focusing on situations where the source is stationary and the observer is in motion.

Explanation of Doppler Shift

1. Defining the Scenario: Let’s say we have a source (s) at rest, which produces a sound of a constant frequency (f₀​). This sound has a wave period (T₀​). An observer is moving away from the source with velocity (v_o​).
2. Observer’s Movement and Crest Detection: At time t = 0, the source produces a crest, which is a high point in the wave. The observer carries a device to count the number of these crests. If the distance between the source and observer is ‘L’ and the speed of sound is ‘v’, the time taken by the observer to detect the first crest (t₁​) can be calculated as L/v.
3. Second Crest and Observer’s Movement: The source produces a second crest after a time period T₀​. During this time, the observer moves a distance of v_o​T₀​. The time taken by the observer to detect the second crest (t₂​) can be found by the equation $$T_0 + (L + T_0v_o)/v$$
4. Detection of Subsequent Crests: If the source produces the n + 1th crest after a time nT, the time taken by the observer to detect this crest can be calculated with a similar formula.
5. Time Gap to Record Crests: The total time to record n crests is the difference between the times it takes to detect the n + 1th crest and the first crest.
6. Time Period of Wave Recorded by Observer: The time period of the wave recorded by the observer (T) is the total time (t) divided by the number of crests (n). This gives $$T = T_0(1 + v_o/v)$$

Derivation of Apparent Frequency

Apparent Frequency (f) is the reciprocal of the time period (T). By manipulating the above equation for T, we can find the apparent frequency when the observer is moving away from the source. After expanding and approximating, we get the equation: $$f = f_0(1 – v_o/v)$$

When the observer moves towards the source, the sign of v_o​ changes, and the apparent frequency becomes $$f = f_0(1 + v_o/v)$$

Summary

In summary, the Doppler Shift explains why the frequency of sound appears to change when the observer is moving relative to the source. It’s an essential principle in various scientific and technological applications, providing insights into the dynamic interaction between waves and moving observers.