Unit 6: Wave Motion

1. Progressive Waves

Waves are disturbances that travel through a medium, transferring energy without the net transfer of matter. A progressive wave, also known as a traveling wave, is a wave that propagates through a medium, continuously carrying energy from one point to another.

Types of Progressive Waves

Progressive waves can be classified based on the direction of vibration of the particles of the medium relative to the direction of wave propagation.

Transverse Waves

A transverse wave is a type of progressive wave in which the particles of the medium vibrate perpendicular to the direction of wave propagation.

Imagine a rope tied to a wall. If you flick the free end up and down, a wave travels along the rope. The segments of the rope move up and down, but the wave itself moves horizontally along the rope. This is a classic example of a transverse wave.

• Characteristics: They consist of crests (points of maximum upward displacement) and troughs (points of maximum downward displacement).
• Examples: Waves on a stretched string, light waves (electromagnetic waves), ripples on the surface of water.

Figure 1: A diagram illustrating a transverse wave would show particles oscillating vertically (perpendicular to the direction of wave propagation), while the wave itself moves horizontally. Crests and troughs would be clearly marked, along with a representation of one wavelength (λ).

Longitudinal Waves

A longitudinal wave is a type of progressive wave in which the particles of the medium vibrate parallel to the direction of wave propagation.

Consider a Slinky spring. If you push one end of the Slinky, a compression travels along its length. The coils of the Slinky move back and forth in the same direction that the compression is traveling. This is a longitudinal wave.

• Characteristics: They consist of compressions (regions where particles are crowded together, high density and pressure) and rarefactions (regions where particles are spread apart, low density and pressure).
• Examples: Sound waves, seismic P-waves.

Figure 2: A diagram illustrating a longitudinal wave would show regions of compression (particles close together) and rarefaction (particles spread apart), with particle displacement occurring parallel to the direction of wave propagation.

Key Characteristics of Progressive Waves

Regardless of whether a wave is transverse or longitudinal, several key parameters describe its properties:

• Amplitude (A):

  The amplitude is the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.

  For a transverse wave, it's the height of a crest or depth of a trough from the equilibrium line. For a longitudinal wave, it relates to the maximum change in pressure or density from the equilibrium state.

• Wavelength (λ):

  The wavelength is the spatial period of a periodic wave, the distance over which the wave's shape repeats. It is the distance between two consecutive corresponding points on the wave, such as two successive crests, troughs, or compressions.

  It is typically measured in meters (m).

• Frequency (f):

  The frequency of a wave is the number of complete oscillations or cycles that pass a given point per unit time.

  It is measured in Hertz (Hz), where 1 Hz = 1 cycle per second.

• Period (T):

  The period of a wave is the time taken for one complete oscillation or cycle to pass a given point.

  It is the reciprocal of frequency: T = 1/f. It is measured in seconds (s).

• Wave Speed (v):

  The wave speed is the speed at which the wave propagates through the medium. It is the distance traveled by a wave per unit time.

  It is related to frequency and wavelength by the fundamental wave equation: v = fλ. It is measured in meters per second (m/s).

2. Mathematical Description of a Wave

To precisely describe a progressive wave, we use mathematical equations that relate its displacement to position and time. For a simple harmonic progressive wave, the displacement y of a particle in the medium at position x and time t can be described by a sinusoidal function.

The General Wave Equation

A common form for a wave traveling in the positive x-direction is:

y(x, t) = A sin(kx - ωt + φ)

Where:

• y(x, t) is the displacement of the particle at position x and time t.
• A is the amplitude, representing the maximum displacement from equilibrium.
• k is the wave number.
• x is the position along the direction of wave propagation.
• ω (omega) is the angular frequency.
• t is the time.
• φ (phi) is the initial phase constant, which determines the displacement at x=0, t=0. Often, for simplicity, we assume φ = 0.

Thus, for a wave traveling in the +x direction, the equation is often simplified to:

y = A sin(kx - ωt)

For a wave traveling in the -x direction, the equation is:

y = A sin(kx + ωt)

Key Parameters in the Wave Equation

Wave Number (k)

The wave number (k), also known as propagation constant, is a measure of the spatial frequency of a wave, representing the number of radians of phase per unit distance.

It is defined as:

k = 2π / λ

Where λ is the wavelength. The unit of wave number is radians per meter (rad/m) or m^-1.

Angular Frequency (ω)

The angular frequency (ω) is a measure of the rate of change of the phase of a sinusoidal wave, representing the number of radians of phase per unit time.

It is defined as:

ω = 2πf

Where f is the frequency. The unit of angular frequency is radians per second (rad/s) or s^-1.

Phase (kx - ωt)

The term (kx - ωt) (or kx + ωt) is called the phase of the wave. It determines the state of oscillation of a particle at a given position and time. Points on a wave that have the same phase are in the same state of oscillation (e.g., both at a crest, or both at an equilibrium position moving in the same direction).

Wave Speed Formulas

The speed of a wave can be expressed in terms of its frequency and wavelength, or its angular frequency and wave number.

1. Using frequency and wavelength:

   v = fλ

   This fundamental relation states that wave speed is the product of its frequency and wavelength.

2. Using angular frequency and wave number:

   Since f = ω / (2π) and λ = 2π / k, we can substitute these into the first formula:

   v = (ω / (2π)) * (2π / k)

   Which simplifies to:

   v = ω / k

   This formula is particularly useful when the wave equation is given in terms of k and ω.

Phase and Phase Difference

The phase of a wave at a point (x, t) is given by (kx - ωt + φ). The phase changes with both position and time.

• Phase Difference between two points at the same time:

  For two points x1 and x2 at the same time t, the phase difference ΔΦ is:

  ΔΦ = (kx2 - ωt) - (kx1 - ωt) = k(x2 - x1)

  If x2 - x1 = λ, then ΔΦ = kλ = (2π/λ)λ = 2π radians. This means points separated by one wavelength are in phase.

• Phase Difference between a point at two different times:

  For a point x at times t1 and t2, the phase difference ΔΦ is:

  ΔΦ = (kx - ωt2) - (kx - ωt1) = -ω(t2 - t1)

  If t2 - t1 = T, then ΔΦ = -ωT = -(2πf)(1/f) = -2π radians. This means the phase repeats after one period.

Example Problem

A progressive wave is described by the equation y = 0.04 sin(20x - 400t), where y and x are in meters and t is in seconds. Determine the amplitude, wave number, angular frequency, wavelength, frequency, and wave speed.

Solution:

Comparing y = 0.04 sin(20x - 400t) with the general equation y = A sin(kx - ωt):

• Amplitude (A): A = 0.04 m
• Wave Number (k): k = 20 rad/m
• Angular Frequency (ω): ω = 400 rad/s
• Wavelength (λ):

  k = 2π / λ &implies; λ = 2π / k = 2π / 20 = π / 10 ≈ 0.314 m

• Frequency (f):

  ω = 2πf &implies; f = ω / (2π) = 400 / (2π) = 200 / π ≈ 63.66 Hz

• Wave Speed (v):

  Using v = ω / k = 400 / 20 = 20 m/s

  Alternatively, using v = fλ = (200 / π) * (π / 10) = 20 m/s

3. Stationary Waves

Unlike progressive waves that transfer energy, stationary waves (also known as standing waves) are characterized by points where the medium is always at rest and points where the medium oscillates with maximum amplitude. They do not appear to travel through the medium.

Formation of Stationary Waves

Stationary waves are formed when two identical progressive waves (having the same amplitude, frequency, and wavelength) travel in opposite directions and superpose upon each other.

The principle of superposition states that when two or more waves overlap, the resultant displacement at any point and at any instant is the vector sum of the displacements due to individual waves at that point and instant.

A common scenario for stationary wave formation is when a progressive wave reflects off a boundary, and the incident wave interferes with the reflected wave.

Nodes and Antinodes

The interference between the two oppositely traveling waves leads to specific points in the medium with distinct behaviors:

Nodes

Nodes are points in a stationary wave where the displacement of the medium particles is always zero. At these points, destructive interference occurs continuously.

Particles at nodes remain stationary, never moving from their equilibrium position.

Antinodes

Antinodes are points in a stationary wave where the displacement of the medium particles is maximum. At these points, constructive interference occurs continuously.

Particles at antinodes oscillate with the maximum possible amplitude, which is twice the amplitude of the individual progressive waves.

Spatial Relationship of Nodes and Antinodes

• The distance between two consecutive nodes is λ/2.
• The distance between two consecutive antinodes is λ/2.
• The distance between a node and an adjacent antinode is λ/4.

Figure 3: A diagram of a stationary wave on a string would show fixed nodes at the ends and oscillating antinodes in between. The positions of nodes and antinodes would be clearly labeled, illustrating the λ/2 and λ/4 relationships.

Mathematical Description of Stationary Waves

Let's consider two identical progressive waves traveling in opposite directions:

Wave 1 (traveling in +x direction): y1 = A sin(kx - ωt)

Wave 2 (traveling in -x direction): y2 = A sin(kx + ωt)

According to the superposition principle, the resultant displacement y is the sum of y1 and y2:

y = y1 + y2 = A sin(kx - ωt) + A sin(kx + ωt)

Using the trigonometric identity sin(P) + sin(Q) = 2 sin((P+Q)/2) cos((P-Q)/2):

Let P = kx - ωt and Q = kx + ωt

(P+Q)/2 = (kx - ωt + kx + ωt) / 2 = 2kx / 2 = kx

(P-Q)/2 = (kx - ωt - (kx + ωt)) / 2 = (-2ωt) / 2 = -ωt

So, the resultant equation for a stationary wave is:

y = 2A sin(kx) cos(-ωt)

Since cos(-θ) = cos(θ):

y = (2A sin(kx)) cos(ωt)

In this equation:

• The term (2A sin(kx)) represents the amplitude of oscillation for a particle at position x. This amplitude varies with position.
• The term cos(ωt) represents the simple harmonic motion in time for all particles. All particles (except at nodes) oscillate with the same angular frequency ω.

Nodes: Occur when sin(kx) = 0. This happens when kx = nπ, where n = 0, 1, 2, .... So, x = nπ/k = nπ / (2π/λ) = nλ/2. Nodes occur at x = 0, λ/2, λ, 3λ/2, ...

Antinodes: Occur when sin(kx) = ±1. This happens when kx = (n + 1/2)π, where n = 0, 1, 2, .... So, x = (n + 1/2)π/k = (n + 1/2)λ/2 = (2n+1)λ/4. Antinodes occur at x = λ/4, 3λ/4, 5λ/4, ...

Characteristics of Stationary Waves

• No Energy Transfer: Unlike progressive waves, stationary waves do not transfer energy across the medium. Energy is confined and oscillates between kinetic and potential forms within segments between nodes.
• Varying Amplitude: The amplitude of oscillation is not constant throughout the medium; it varies from zero at nodes to maximum at antinodes.
• Phase Relationships: All particles between two consecutive nodes oscillate in phase with each other. However, particles on either side of a node oscillate 180° out of phase with each other.
• Fixed Positions: Nodes and antinodes occur at fixed positions in the medium.

Examples of Stationary Waves

Stationary Waves on a Stretched String

When a string is fixed at both ends and vibrated, standing waves can be formed. The fixed ends must always be nodes.

• Boundary Conditions: Both ends of the string must be nodes.
• Harmonics and Overtones:
  • Fundamental Mode (First Harmonic, n=1): The simplest standing wave pattern, with one antinode in the middle and nodes at both ends. The length of the string L is equal to λ1/2.

    L = λ1/2 &implies; λ1 = 2L

    The fundamental frequency is f1 = v / λ1 = v / (2L)

  • Second Harmonic (First Overtone, n=2): Two antinodes and three nodes. The length of the string L is equal to λ2.

    L = λ2 &implies; λ2 = L

    The frequency is f2 = v / λ2 = v / L = 2(v / (2L)) = 2f1

  • Third Harmonic (Second Overtone, n=3): Three antinodes and four nodes. The length of the string L is equal to 3λ3/2.

    L = 3λ3/2 &implies; λ3 = 2L/3

    The frequency is f3 = v / λ3 = v / (2L/3) = 3(v / (2L)) = 3f1

• General Formula for Frequencies: For the n^th harmonic, the frequency is fn = n * f1 = n * (v / (2L)), where n = 1, 2, 3, ...

Figure 4: Diagrams illustrating the first three harmonics on a stretched string fixed at both ends, showing the positions of nodes and antinodes for each mode of vibration.

Stationary Waves in Air Columns

Standing waves can also be formed in air columns, such as in organ pipes or wind instruments. The boundary conditions for air columns are different:

• An open end of a pipe is an antinode (maximum displacement of air particles, minimum pressure variation).
• A closed end of a pipe is a node (minimum displacement of air particles, maximum pressure variation).

Open Organ Pipes (Open at both ends)

• Boundary Conditions: Antinodes at both ends.
• Harmonics and Overtones:
  • Fundamental Mode (First Harmonic, n=1): An antinode at each end and one node in the middle. The length of the pipe L is equal to λ1/2.

    L = λ1/2 &implies; λ1 = 2L

    The fundamental frequency is f1 = v / λ1 = v / (2L)

  • Second Harmonic (First Overtone, n=2): Two nodes and three antinodes. The length of the pipe L is equal to λ2.

    L = λ2 &implies; λ2 = L

    The frequency is f2 = v / λ2 = v / L = 2f1

• General Formula for Frequencies: For the n^th harmonic, the frequency is fn = n * f1 = n * (v / (2L)), where n = 1, 2, 3, ... (All harmonics are present).

Figure 5: Diagrams illustrating the first three harmonics in an open organ pipe, showing antinodes at both ends and varying numbers of nodes in between.

Closed Organ Pipes (Closed at one end, open at the other)

• Boundary Conditions: A node at the closed end and an antinode at the open end.
• Harmonics and Overtones:
  • Fundamental Mode (First Harmonic, n=1): One node at the closed end and one antinode at the open end. The length of the pipe L is equal to λ1/4.

    L = λ1/4 &implies; λ1 = 4L

    The fundamental frequency is f1 = v / λ1 = v / (4L)

  • First Overtone (Third Harmonic, n=3): Two nodes and two antinodes. The length of the pipe L is equal to 3λ3/4.

    L = 3λ3/4 &implies; λ3 = 4L/3

    The frequency is f3 = v / λ3 = v / (4L/3) = 3(v / (4L)) = 3f1

  • Second Overtone (Fifth Harmonic, n=5): Three nodes and three antinodes. The length of the pipe L is equal to 5λ5/4.

    L = 5λ5/4 &implies; λ5 = 4L/5

    The frequency is f5 = v / λ5 = v / (4L/5) = 5(v / (4L)) = 5f1

• General Formula for Frequencies: For the n^th harmonic, the frequency is fn = n * f1 = n * (v / (4L)), where n = 1, 3, 5, ... (Only odd harmonics are present).

Figure 6: Diagrams illustrating the first three possible harmonics (fundamental, first overtone, second overtone) in a closed organ pipe, showing a node at the closed end and an antinode at the open end for each mode.

Summary Table: Comparison of Wave Types