Introduction to Waves Overview

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Questions and Answers

In a medium where the bulk modulus is increased, while the density remains constant, how does the speed of sound in that medium change?

  • The speed of sound changes unpredictably.
  • The speed of sound decreases.
  • The speed of sound increases. (correct)
  • The speed of sound remains constant.

Two waves of the same frequency and amplitude traveling in opposite directions on a string interfere. What is the resulting wave pattern called?

  • Standing wave (correct)
  • Transverse wave
  • Progressive wave
  • Longitudinal wave

Why is the speed of sound in solids and liquids generally greater than in gases despite their higher density?

  • Solids and liquids are more elastic than gases. (correct)
  • Solids and liquids have lower bulk moduli than gases.
  • Solids and liquids have higher compressibilities than gases.
  • Solids and liquids have weaker intermolecular forces than gases.

What is the physical significance of the adiabatic index (γ) in the formula for the speed of sound in a gas?

<p>It reflects the relationship between the specific heat capacities of the gas at constant pressure and constant volume. (A)</p> Signup and view all the answers

Consider two identical strings under tension. String A has a linear mass density (μ) that is twice that of String B. What is the ratio of the wave speeds on String A to String B?

<p>1:√2 (C)</p> Signup and view all the answers

A vibrating string produces a wave described by the equation y(x,t) = A sin(kx - ωt). What is the wave's speed?

<p>ω/k (C)</p> Signup and view all the answers

What will be the effect on the speed of sound in air if the atmospheric pressure is increased while keeping the temperature constant?

<p>The speed of sound will increase. (A)</p> Signup and view all the answers

A guitar string is plucked, creating a standing wave pattern. The string is then tightened. What is the resulting change in the frequency of the fundamental mode of vibration?

<p>The fundamental frequency increases. (B)</p> Signup and view all the answers

What is the correct formula for the wavelength $oxed{ ext{λ}}$ in terms of the length $oxed{L}$ and the harmonic number $oxed{n}$?

<p>$ rac{2L}{n}$ (A)</p> Signup and view all the answers

Which frequency corresponds to the 2nd harmonic of a wave in terms of the string length $L$ and wave speed $v$?

<p>$ rac{2v}{L}$ (D)</p> Signup and view all the answers

What is the phenomenon called when two waves of nearly equal frequencies interact?

<p>Beats (C)</p> Signup and view all the answers

When two waves undergo constructive interference, what is the outcome related to their amplitudes?

<p>They amplify each other (A)</p> Signup and view all the answers

What is the term for the natural frequencies of vibration of a completed system?

<p>Normal modes (C)</p> Signup and view all the answers

How is the speed of a mechanical wave primarily determined?

<p>By the inertial and elastic properties of the medium (C)</p> Signup and view all the answers

What phenomenon describes the rise and fall of sound intensity when two waves interact?

<p>Waxing and waning (D)</p> Signup and view all the answers

Which type of wave has particle oscillation perpendicular to the direction of wave propagation?

<p>Transverse waves (B)</p> Signup and view all the answers

What phenomenon explains why a leaf floating on disturbed water moves up and down without moving towards the shore?

<p>Wave propagation without medium transfer (C)</p> Signup and view all the answers

Which of the following statements is true regarding mechanical waves?

<p>They require a medium for propagation. (A)</p> Signup and view all the answers

In what manner does a sound wave propagate through air?

<p>By compressions and rarefactions (A)</p> Signup and view all the answers

Which type of wave cannot travel through a vacuum?

<p>Mechanical waves (A)</p> Signup and view all the answers

The principle of superposition of waves describes what phenomenon?

<p>The interference of overlapping waves (B)</p> Signup and view all the answers

How does the speed of a longitudinal wave compare to that of a transverse wave in the same medium?

<p>There is no fixed relationship; it depends on the medium (D)</p> Signup and view all the answers

What characteristic is unique to an electromagnetic wave compared to mechanical waves?

<p>They propagate through vacuums (B)</p> Signup and view all the answers

What occurs when two waves of slightly different frequencies interfere with each other?

<p>Beats are produced, creating a fluctuating amplitude (B)</p> Signup and view all the answers

What is the fundamental frequency of a closed organ pipe characterized by n = 1?

<p>The first harmonic (C)</p> Signup and view all the answers

In a closed organ pipe, how many harmonics are present?

<p>Only odd harmonics (A)</p> Signup and view all the answers

Which equation accurately describes the position of nodes in a closed organ pipe?

<p>$x = n \frac{\lambda}{2}$ (C)</p> Signup and view all the answers

For a closed organ pipe of length L, what is the equation for the wavelength?

<p>$\lambda = (2n-1) \frac{4L}{n}$ (D)</p> Signup and view all the answers

In the case of an open organ pipe, what type of wave formation occurs at the ends?

<p>Antinodes at both ends (C)</p> Signup and view all the answers

Which of the following statements about the reflected wave in an open organ pipe is true?

<p>It is in phase with the incident wave. (A)</p> Signup and view all the answers

What is the frequency formula for a given harmonic n in a closed organ pipe?

<p>$v_n = (2n-1)v_1$ (C)</p> Signup and view all the answers

Which expression represents the resultant wave in terms of the initial and reflected waves in an open organ pipe?

<p>$y(x,t) = 2a cos(kx) cos(wt)$ (B)</p> Signup and view all the answers

What is the result of constructive interference between two waves?

<p>The resultant wave has an amplitude equal to the sum of the individual amplitudes. (A)</p> Signup and view all the answers

When does destructive interference occur between two waves?

<p>When the phase difference is $ ext{π}$ or an odd multiple of $ ext{π}$. (B)</p> Signup and view all the answers

What happens to a wave reflected from a rigid boundary?

<p>The reflected wave suffers a phase change of $180^ ext{°}$. (D)</p> Signup and view all the answers

In a standing wave formed on a string fixed at both ends, what are the nodes?

<p>Positions where the amplitude is zero. (D)</p> Signup and view all the answers

What characterizes the natural frequencies of a vibrating string fixed at both ends?

<p>They are influenced by the boundary conditions at the ends of the string. (B)</p> Signup and view all the answers

How does the wave speed in the equation $v_n = \frac{n v}{2L}$ relate to the natural frequencies?

<p>It varies directly with the harmonic number $n$. (A)</p> Signup and view all the answers

Which statement about the behavior of waves upon reflection at a free end is correct?

<p>The reflected wave has the same phase and amplitude as the incident wave. (B)</p> Signup and view all the answers

What defines the antinodes in a standing wave on a string?

<p>Regions of maximum amplitude. (A)</p> Signup and view all the answers

Which of the following equations correctly represents a wave in a medium where 'y' is the wave function, 'v' is the wave speed, 'x' is the spatial coordinate, and 't' is time?

<p>$\frac{\partial ^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial ^2 y}{\partial t^2}$ (C)</p> Signup and view all the answers

Which of the following statements accurately describes how the speed of a transverse wave on a stretched string is affected by the string's properties?

<p>The speed of the wave is directly proportional to the square root of the tension and inversely proportional to the square root of the linear mass density. (C)</p> Signup and view all the answers

A transverse wave travels along a rope. If the rope's tension is increased, what happens to the speed of the wave?

<p>The wave speed increases. (B)</p> Signup and view all the answers

Why is the speed of a wave different for different types of waves (longitudinal vs. transverse) in the same medium?

<p>The speed of a wave is determined by the elasticity of the medium, and the elasticity of the medium is different for different types of waves. (C)</p> Signup and view all the answers

Which of the following accurately describes the motion of particles in a longitudinal wave?

<p>Particles oscillate parallel to the direction of wave propagation. (D)</p> Signup and view all the answers

A harmonic wave is created on a rope by giving it continuous periodic up and down jerks. Which of the following statements describes the motion of the particles in the rope?

<p>The particles oscillate perpendicular to the direction of wave propagation, the rope itself doesn't move. (C)</p> Signup and view all the answers

Which of the following correctly describes why a single pulse traveling along a rope is considered a traveling wave?

<p>The pulse travels with a constant speed, and the rope's particles oscillate perpendicular to the direction of the pulse's movement. (D)</p> Signup and view all the answers

What is the primary reason for the difference in speed between a longitudinal wave and a transverse wave within the same medium?

<p>The difference in wave speed arises from the varying elasticity of the medium with respect to compression and shear forces. (B)</p> Signup and view all the answers

Flashcards

Wave

A pattern that transmits energy through a medium without transferring matter.

Transverse Wave

A wave where particle displacement is perpendicular to the wave direction.

Longitudinal Wave

A wave where particle displacement is parallel to the wave direction.

Mechanical Wave

A wave that requires a medium to propagate, like sound or water waves.

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Electromagnetic Wave

A wave that can travel through a vacuum, like light or radio waves.

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Matter Wave

Waves associated with particles of matter, like electrons and atoms.

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Principle of Superposition

When two waves meet, the resulting wave is the sum of their displacements.

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Compressions and Rarefactions

Regions of high pressure (compressions) and low pressure (rarefactions) in a longitudinal wave.

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Constructive Interference

Occurs when waves add together, increasing amplitude.

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Destructive Interference

Occurs when waves cancel each other, reducing amplitude.

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Phase Difference

The difference in phase between two waves, affecting interference.

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Reflection at Rigid Boundary

Wave reflects with a 180° phase change at a rigid boundary.

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Reflection at Free End

Wave reflects without a phase change at a free end.

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Standing Waves

Result from two waves moving in opposite directions, creating nodes and antinodes.

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Nodes

Points of zero amplitude in a standing wave.

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Normal Modes

Natural frequencies of vibration determined by boundary conditions.

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Fundamental Mode

The first harmonic of vibration, corresponding to n = 1.

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Harmonics

Overtones are frequencies at integer multiples of the fundamental frequency.

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Closed Organ Pipe

A pipe closed at one end where only odd harmonics are present.

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Position of Nodes

Positions in a standing wave where there is no displacement, x = nλ/2.

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Position of Antinodes

Positions of maximum displacement in a wave, x = (2n-1)λ/4.

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Wavelength in Closed Pipe

The formula for wavelength is λ = (2n-1)(4L/n).

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Frequencies in Closed Pipe

Frequencies are given as v_n = (2n-1)v_1.

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Open Organ Pipe

A pipe open at both ends where antinodes are formed at both ends.

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Traveling Wave

A single pulse that moves through a medium at a constant speed.

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Harmonic Wave

A wave produced by continuous periodic disturbances.

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Speed of a Wave

Depends on the inertial and elastic properties of a medium.

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Wave Function

Mathematical representation of a wave's properties in space and time.

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Wave Equation

Equation relating wave displacement to speed in medium: $ rac{oldsymbol{ ext{∂}}^2 y}{oldsymbol{ ext{∂}} x^2} = rac{1}{v^2} rac{oldsymbol{ ext{∂}}^2 y}{oldsymbol{ ext{∂}} t^2}$.

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Transverse Wave Speed

Speed of a transverse wave on a stretched string depends on linear mass density and tension.

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Antinodes position

The positions of maximum displacement in standing waves, given by kx = nπ.

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Wavelength formula

Wavelength λ of a wave in terms of length L and harmonic number n: λ = 2L/n.

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Beats

The interference pattern produced by two waves of similar frequencies resulting in amplitude variations.

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Interference

The phenomenon where two or more waves overlap, resulting in a new wave pattern.

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Speed of Transverse Waves

The speed of transverse waves on a stretched string is given by v = √(T/μ), where T is tension and μ is linear density.

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Speed of Longitudinal Waves

The speed of longitudinal waves, such as sound, is given by v = √(B/p), where B is the bulk modulus and p is the density.

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Speed of Sound in Gases

The speed of sound in a gas is given by v = √(γP/p), where γ is the adiabatic index, P is pressure, and p is density.

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Newton's Formula

Newton's formula assumes that sound propagation in a gas keeps pressure and volume constant.

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Adiabatic Process

An adiabatic process occurs when PV^γ = Constant, where γ is the ratio of specific heats.

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Wave Interference

When two waves of the same frequency and amplitude overlap, they create a new pattern with varied amplitude.

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Resultant Amplitude

The resultant amplitude during interference can be greater, less, or zero compared to individual wave amplitudes.

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Study Notes

Introduction to Waves

  • Waves are a phenomenon where energy is transferred from one location to another without the actual transfer of matter.
  • Examples include dropping a stone in a pond, or the motion of springs connected together.

Propagation of Sound Waves

  • Sound waves propagate through air by compressions and expansions of air molecules.
  • Compressions are regions of higher density and pressure, while rarefactions are regions of lower density and pressure.
  • These compressions and rarefactions create pressure variations which propagate as the wave moves further.

Types of Mechanical Waves

  • Transverse waves: In these waves, the oscillations are perpendicular to the direction of wave propagation.
    • Example: Waves on a stretched string
  • Longitudinal waves: In these waves, the oscillations are along the direction of wave propagation.
    • Example: Sound waves in air

Speed of Mechanical Waves

  • The speed of a mechanical wave depends on the characteristics of the medium through which it travels.
  • Inertial property (e.g., mass density) and elastic properties of the medium (e.g., bulk modulus) affect the speed.
  • The speed of a transverse wave on a stretched string is determined by the tension (T) and the linear mass density (μ) of the string: v = √(T/μ). Speed of longitudinal wave is given by v=√(B/p) where B is the bulk modulus and ρ is the density of the medium.
  • Different media will have different speeds for both transverse and longitudinal waves.
  • Newton's formula was introduced to estimate the speed of sound in air, considering isothermal compression and expansion.
  • Laplace's correction improves the estimate by considering adiabatic expansion and compression, resulting in a more accurate speed calculation.

Superposition of Waves

  • When two pulses travelling in opposite directions overlap, their displacements add up according to the principle of superposition.
  • Constructive interference occurs when the displacements of the waves add up to increase the amplitude.
  • Destructive interference occurs when the displacements of the waves add up to decrease the amplitude.

Standing Waves

  • Stationary waves result from the superposition of two waves travelling in opposite directions.
  • These waves exhibit nodes (points of zero displacement) and antinodes (points of maximum displacement).
  • The resulting wave remains stationary in space as opposed to travelling waves.

Normal Modes in Pipes

  • Normal modes refer to the specific frequencies at which a wave can establish a standing wave pattern within a confined space like a pipe.
  • In an open pipe, antinodes are formed at both ends, while in a closed pipe, a node is formed at the closed end and an antinode at the open end.
  • These frequencies depend on the length of the pipe, speed of the wave and the boundary conditions Odd harmonics are present in a closed pipe, while both odd and even harmonics are present in an open pipe

Beats

  • Beats are the periodic variations in the amplitude of sound produced by the superposition of two sound waves with slightly different frequencies.
  • The beat frequency is equal to the difference between the frequencies of the two waves.

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