Brief Summary
This video provides a comprehensive overview of waves, covering both string waves and sound waves. It begins with the definition of a wave as a transfer of information and progresses through various concepts, equations, and problem-solving techniques. The video includes numerous examples and past year questions (PYQs) to help viewers master the topic.
- Definition and types of waves
- Wave equation and mathematical representation
- Superposition and interference of waves
- Standing waves and resonance
- Sound waves and their properties
Definition of a Wave
A wave is defined as the transfer of information. In the context of a string wave, if one end of a string is jerked, a pulse is created, and this pulse travels forward, transferring information from one particle of the string to another. The particles themselves only move vertically, perpendicular to the x-axis, and the y-coordinate is the information being transferred. For example, if a wave has a speed of 1 metre per second, a y-coordinate of 2 cm at x = 0 will travel 1 metre in 1 second, meaning the particle at x = 1 will then possess that y-coordinate.
Transverse Waves and Wave Equation
String waves, where particles travel perpendicular to the wave's propagation, are called transverse waves. Mathematically, if the y-coordinate of a particle at x = 0 is a function of time, f(t), then at a general point x, the y-coordinate at time t is f(t - x/v), where v is the wave speed. Any function of the form t - x/v represents a travelling wave. For instance, y = a sin(ωt - ωx/v) is a travelling wave. Similarly, any function of x - vt also represents a travelling wave. A wave travelling in the negative x direction is represented by t + x/v or x + vt.
JEE Question on Travelling Waves
A question from JEE 2021 asks which of the given options represents a travelling wave. The correct answer is the one where t and x are grouped together in the form of x - vt or t - x/v. For example, if a wave travels in the positive x direction and at t = 0, y = 1/(1 + x²), then at a general time t, y = 1/(1 + (x - vt)²).
Sinusoidal Waves and Wave Properties
For a sinusoidal wave, where the particle at the origin performs simple harmonic motion (SHM), the y-coordinate at a general x is a sin(ωt - kx + φ), where k = ω/v. The wave number k is equal to 2π/λ, where λ is the wavelength. Wavelength is the distance travelled by the wave in one oscillation. The velocity of a particle at some time t is dy/dt, keeping x constant, which equals ωa cos(ωt - kx + φ). The slope of the wave is dy/dx, keeping time constant, which equals -ka cos(ωt - kx + φ). The relationship between the particle's velocity and the slope is given by the particle velocity = -v * slope, where v is the wave velocity.
JEE Questions on Wave Equations
A question from JEE 2024 provides a progressive wave equation and asks for the frequency. By comparing the given equation to a sin(ωt - kx + φ), the frequency can be determined. Another question from JEE 2023 gives a wave equation and asks for the wave speed, requiring careful attention to units. A question from JEE 2025 provides a sinusoidal wave equation and asks for the velocity of the wave, again emphasising the importance of unit consistency.
Wave Equation and Properties
A question from JEE 2025 provides a sinusoidal wave and asks to determine the equation representing the wave. By using the information given, such as the wavelength and the wave's behaviour at t = 0 and x = 0, one can eliminate options and find the correct equation. A question from JEE 2019 asks for the distance between two consecutive points with a phase difference of 60 degrees in a wave of frequency 500 Hz.
Wave Speed in a String
The speed of a travelling wave in a string is given by v = √(T/μ), where T is the tension in the string and μ is the mass per unit length. A question from JEE 2020 provides the initial tension and speed, then changes the tension and asks for the new tension. A question from JEE 2019 involves a car accelerating and changing the wave speed in a string connected to a ball, requiring the determination of the car's acceleration using the relationship between tension and wave speed.
Wave Speed and Material Properties
A question from JEE 2025 involves two strings with circular cross-sections made of the same material and stretched to the same tension. The ratio of the velocities of transverse waves in the two strings is asked, requiring the understanding that the density is the same but the linear mass density differs due to different radii.
Interference of Waves
When two waves, y1 = a1 sin(ωt - kx + φ1) and y2 = a2 sin(ωt - kx + φ2), are simultaneously present in a string, they interfere. The resultant y-coordinate of the string is y1 + y2. The concept of phasors is introduced to elegantly calculate the resultant wave. The resultant amplitude is given by a_net = √(a1² + a2² + 2a1a2 cos θ), where θ is the phase difference. The phase angle α of the resultant wave is given by tan α = (a2 sin θ) / (a1 + a2 cos θ).
JEE Questions on Wave Interference
A question from JEE 2023 involves two waves with equal amplitudes interfering, resulting in the same amplitude. The phase difference between the waves is asked. A question from JEE 2022 provides two wave equations and asks for the resultant amplitude, requiring the calculation of the phase difference and the application of the amplitude formula. A question from JEE 2020 involves three waves with different phase angles superposing, and the resultant intensity is asked, requiring the understanding that intensity is proportional to the square of the amplitude.
Wave Interference and Phase Constant
A question from 2025 provides two wave equations and asks for the amplitude and phase constant of the resultant wave. The amplitude is calculated using the formula a_net = √(a1² + a2² + 2a1a2 cos θ), and the phase constant is calculated using tan α = (a2 sin δ) / (a1 + a2 cos δ).
Standing Waves
Standing waves occur when an incident wave gets reflected from a wall, resulting in a phase change of π. The resultant y-coordinate of the string is given by y = 2a sin(kx) cos(ωt). Points where the y-coordinate is always zero are called nodes, and they occur when kx = nπ, or x = nλ/2. The points where the amplitude is maximum are called antinodes, and they are located exactly between the nodes.
Resonant Modes and Harmonics
Standing waves are created in a string connected between two walls. The possible modes are determined by the condition that the end points must be nodes. The length of the string is given by L = nλ/2, where n is the number of loops. The possible frequencies are given by f = nv/2L, where v = √(T/μ). The fundamental mode (n = 1) is the smallest frequency that can produce a standing wave. Higher modes are called overtones and harmonics.
Sonometer and Wave Properties
A sonometer is used to experimentally set up standing waves in a wire. The wire is connected to a block over a pulley, and the tension in the wire is equal to the weight of the block. The frequency of the standing wave is given by f = nv/2L, where v = √(mg/μ). A question from JEE 2024 involves a sonometer wire with a given resonating length and fundamental frequency, and the new resonating length for a different fundamental frequency is asked.
Resonating Frequencies and Harmonics
A question from JEE 2021 involves a wire connected between two rigid walls with a given linear mass density and tension. The length of the wire is asked, given two consecutive resonating frequencies.
Sound Waves and Longitudinal Nature
Sound waves are longitudinal waves, meaning the oscillations are parallel to the propagation of the waves. The information that gets transferred is the pressure. The pressure wave is given by ΔP = P_not sin(ωt - kx + φ), where P_not is the amplitude of the pressure wave. The speed of a sound wave is given by v = √(B/ρ), where B is the bulk modulus.
Bulk Modulus and Adiabatic Processes
The bulk modulus is defined as B = -ΔP / (ΔV/V). For gases, the sound propagation is adiabatic, meaning there is no energy exchange. In an adiabatic process, PV^γ is constant, and the bulk modulus is given by B = γP. Therefore, the speed of sound in a gas is v = √(γRT/M), where M is the molar mass of the gas in kilograms.
Sound Speed and Molecular Mass
A question from JEE 2023 asks for the ratio of the speed of sound in hydrogen to oxygen at the same temperature. The ratio is determined using the formula v = √(γRT/M), considering that both are diatomic molecules and have the same γ.
Loudness and Intensity
The loudness of a sound is given by L = 10 log(I/I_not), where I is the intensity of the sound wave and I_not is the minimum intensity that can be heard by humans (10^-12 W/m²). The units of loudness are decibels. For a point source emitting sound waves, the intensity at a distance R is given by I = Power / (4πR²).
Sound Intensity and Distance
A question from JEE 2019 involves a small speaker delivering a certain output power, and the distance at which the loudness is 120 decibels is asked. The intensity is calculated using the formula I = Power / (4πR²), and the loudness formula is used to find the distance.
Standing Waves in Sound and Organ Pipes
Standing waves are created in organ pipes. In an open pipe (open at both ends), the pressure is equal to the atmospheric pressure at the open ends, meaning the change in pressure is zero. Therefore, the open ends are pressure nodes. The possible frequencies are given by f = nv/2L, where v = √(γRT/M).
Closed Pipes and Harmonics
In a closed pipe (closed at one end), the closed end is a pressure antinode. The possible frequencies are given by f = (2n - 1)v/4L. Only odd harmonics are present in a closed pipe.
Open vs Closed Pipes
A question from JEE 2024 involves a closed organ pipe and an open organ pipe, and the length of the closed pipe is asked, given that the fundamental frequency of the closed pipe is equal to the first overtone of the open pipe. A question from JEE 2024 involves a closed organ pipe with a certain fundamental frequency, and water is poured into it, changing the fundamental frequency. The mass of water poured is asked.
Closed Pipes and Volume
A closed organ pipe is filled with water up to 1/5 of its volume, and the percentage change in the fundamental frequency is asked.
Harmonics and Frequencies
A closed and open pipe are filled with two gases with the same bulk modulus but different densities. The frequency of the ninth harmonic of the closed tube is equal to that of the fourth harmonic of the open tube, and the length of the open tube is asked.
Resonance Tube Experiment
In a resonance tube experiment, a tube is filled with water, and the water level can be changed. A tuning fork is used to set up standing waves. The air column behaves like a closed pipe. The node is not exactly at the open end but a little above it, and this distance is called the end correction (E ≈ 0.6 * radius). The frequency is given by f = v / (4(L + E)).
Resonance and Wavelength
A question from JEE 2020 involves a tuning fork in resonance with a tube filled with water. The water level is raised, and the next resonance is achieved. The speed of sound is asked. The difference between two consecutive modes is one loop length (λ/2).
Resonance and Diameter
A question from JEE 2021 involves a resonance tube column with a given diameter. A tuning fork is used, and the column length for the first resonance is asked.
Beats and Frequency Difference
If two sound waves have frequencies that differ slightly (f and f + Δf), they interfere and produce beats. The intensity varies with time, and the beat frequency is Δf.
Beat Frequency and Harmonics
A closed organ pipe and an open organ pipe have a beat frequency of 7 Hz. The speed of sound is asked.
Tension and Beat Frequency
A tuning fork resonates with a sonometer wire of length 1 metre stretched with a tension of 6 Newtons in the fundamental mode. When the tension is increased to 54 Newtons, the same tuning fork produces 12 beats per second. The tuning fork frequency is asked.
Air Columns and Beat Frequency
In a resonance tube experiment, air columns of 1 and 1.2 metres give 15 beats per second in the respective fundamental mode. The speed of sound is asked.
