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Interactive physics simulator

Transverse Wave

Observe the dynamics of transverse oscillations. Study wave anatomy, measure crests and troughs using virtual calipers, compare wave speeds on multiple strings under different tension and density, and explore electromagnetic polarization under Malus's Law.

Transverse Wave Laboratory ⓘ

Interact with the sliders to change parameters in real-time. Turn on 'Freeze Frame' to measure with calipers.

Anatomy Lab

Live Wave Telemetry

Wave Speed (v1)
2.00 m/s
Wavelength (λ1)
2.00 m
Frequency (f)
1.00 Hz
Tension (T1)
10.0 N
Polarized Amplitude
100%

Introduction to Transverse Waves

A transverse wave is a wave in which the particles of the medium vibrate or oscillate perpendicularly (at right angles) to the direction that the wave propagates. This is in direct contrast to a longitudinal wave, where particles vibrate parallel to the direction of wave travel.

A classic example of a transverse wave is a wave traveling along a stretched string. When one end is shaken vertically, a wave travels horizontally down the string, but the individual fibers of the string only move up and down, never traveling along with the wave pulse.

The Anatomy of a Transverse Wave

Continuous, periodic transverse waves are described by several geometric and physical parameters:

  • Crest: The point of maximum positive (upward) displacement of a medium particle from its equilibrium position.
  • Trough: The point of maximum negative (downward) displacement from equilibrium.
  • Amplitude (A): The maximum displacement of a particle from its rest position. Amplitude is directly related to the energy carried by the wave: Wave Intensity I ∝ A².
  • Wavelength (λ): The spatial distance between two consecutive identical points on the wave that are in phase, such as from crest to adjacent crest, or trough to adjacent trough.
  • Equilibrium: The flat, undisturbed centerline representing the medium when no wave is present.

Wave Speed on a Stretched String

The speed v at which a transverse wave propagates along a stretched string is determined entirely by the mechanical properties of the string medium. By analyzing the tension and inertia of the string particles, the wave speed is given by:

v = √(T / μ)

Where:

  • T: The tension applied to the string, measured in Newtons (N). Tension acts as the restoring force pulling displaced particles back to equilibrium.
  • μ: The linear mass density of the string, measured in kilograms per meter (kg/m). It represents the inertia of the medium.

Because speed is set strictly by the medium, changing the frequency of the source will not change the wave speed. Instead, if frequency increases, the wavelength adjusts inversely (λ = v/f) to maintain the speed dictated by the string\'s tension and mass density.

The Phenomenon of Wave Polarization

Because particles in a transverse wave oscillate perpendicular to the direction of travel, their displacement can lie in any orientation within a two-dimensional plane. For example, a string wave could vibrate vertically, horizontally, or at a diagonal.

Polarization is the process of restricting these multi-directional vibrations to a single plane of oscillation. Light, being an electromagnetic wave composed of oscillating electric and magnetic fields, is a transverse wave. Standard light from a bulb is unpolarized, vibrating randomly in all directions.

When unpolarized light passes through a polarizing filter, only the component of the wave vibrating parallel to the filter\'s transmission axis is transmitted. If this light then encounters a second polarizer (known as an analyzer) at an angle θ, the transmitted amplitude decays according to Malus\'s Law:

A = A0 · cos(θ)

And the transmitted intensity is:

I = I0 · cos²(θ)

If the analyzer is rotated to θ = 90° (crossed polarizers), the transmitted wave amplitude is zero, and the light is completely blocked. Longitudinal waves (like sound in air) cannot be polarized, making polarization the defining diagnostic test to identify if a wave is transverse.

Solved Examples

Example 1

A transverse wave propagates along a string. The distance between a crest and the adjacent trough is 0.25 meters. If the wave completes 15 cycles in 3.0 seconds, calculate (a) the wavelength and (b) the frequency of the wave.

View Step-by-Step Solution
  1. Identify the given values: distance from crest to adjacent trough is half of a wavelength, and 15 cycles are completed in 3.0 seconds.
  2. Part (a): By definition, the distance from a crest to the very next trough is exactly λ/2.
  3. So, λ/2 = 0.25 m ⇒ λ = 0.50 meters.
  4. Part (b): Frequency is the number of cycles per unit time: f = cycles / time.
  5. Substitute values: f = 15 cycles / 3.0 s = 5.0 Hz.
  6. The wavelength of the wave is 0.50 m and the frequency is 5.0 Hz.

**Final Answer:** λ = 0.50 m, f = 5.0 Hz

Example 2

A guitar string has a length of 0.65 meters and a mass of 4.0 grams (0.004 kg). If the tension in the string is tuned to 160 Newtons, find the speed of a transverse wave traveling along this string.

View Step-by-Step Solution
  1. Identify variables: length L = 0.65 m, mass m = 0.004 kg, tension T = 160 N.
  2. Calculate the linear mass density (μ) of the string: μ = m / L.
  3. Substitute: μ = 0.004 kg / 0.65 m ≈ 0.00615 kg/m.
  4. Use the wave speed formula for a stretched string: v = √(T / μ).
  5. Substitute values: v = √(160 / 0.00615) ≈ √(26000) ≈ 161.2 m/s.
  6. The speed of the transverse wave is approximately 161.2 m/s.

**Final Answer:** v ≈ 161.2 m/s

Example 3

Unpolarized light of amplitude A0 passes through a polarizer to become vertically polarized. It then encounters a second polarizer (analyzer) whose transmission axis is oriented at 60° to the vertical. Calculate (a) the amplitude of the emerging wave in terms of A0, and (b) the percentage of the wave intensity that is transmitted.

View Step-by-Step Solution
  1. Identify the variables: initial amplitude after the first polarizer is A1 = A0 (assuming vertical), angle θ = 60°.
  2. Part (a): According to Malus's Law for amplitude, the transmitted amplitude is A = A1 cos(θ).
  3. Substitute: A = A0 cos(60°) = A0 · 0.5 = 0.5 A0.
  4. Part (b): Wave intensity is proportional to the square of its amplitude: I ∝ A².
  5. Therefore, the ratio of transmitted to incident intensity is I/I0 = (A/A0)² = (0.5)² = 0.25.
  6. Expressed as a percentage: 0.25 · 100% = 25%.
  7. The emerging amplitude is 0.5 A0 and 25% of the intensity is transmitted.

**Final Answer:** A = 0.5 A0, Intensity Transmitted = 25%

Common Misconceptions & Pitfalls

  • Misconception: The speed of a wave on a string increases if you wave the string faster.
    **Reality:** No. Waving the string faster increases the frequency, but the speed stays the same because it only depends on the string\'s tension and thickness. As a result, the wavelength simply shrinks.
  • Misconception: Light waves and sound waves can both be polarized because they are waves.
    **Reality:** Only transverse waves can be polarized. Light is transverse and can be polarized. Sound in fluids is longitudinal and cannot be polarized, as particles only move back and forth along the path of the wave.
  • Misconception: Transverse waves require a material medium.
    **Reality:** Mechanical transverse waves (like string waves) require a physical medium. However, electromagnetic waves (like light and radio waves) are non-mechanical transverse waves that propagate through empty vacuum.

Practice Questions

Question 1

Explain how tension and linear mass density control the propagation speed of a transverse wave on a string.

Show Explanation

The speed of a transverse wave on a string depends on two competing physical properties: tension (T) and linear mass density (μ). Tension acts as the elastic restoring force. When a particle is displaced, a higher tension pulls it back toward equilibrium faster, transferring the mechanical signal to the neighboring particle more rapidly and increasing the wave speed. Conversely, linear mass density represents the mass per unit length (inertia). A higher mass density means adjacent particles are heavier and require more force and time to accelerate, which slows down the rate at which the disturbance propagates.

Question 2

Why is polarization possible only for transverse waves, and why can longitudinal waves not be polarized?

Show Explanation

Polarization involves restricting the direction of oscillation to a single plane. For a transverse wave, particles oscillate perpendicular to the wave propagation axis, allowing for infinitely many vibration angles in a 2D plane (e.g., vertical, horizontal, diagonal). A polarizing filter can select one direction and block the rest. For a longitudinal wave (such as sound in air), the particles oscillate only parallel to the wave propagation direction. Since there is only one possible axis of motion (along the line of travel), there are no perpendicular components to filter or isolate, making polarization physically impossible.

Question 3

Describe what happens to a transverse wave on a string when it reflects from a fixed boundary versus a free boundary.

Show Explanation

When a transverse wave reaches a fixed boundary, the end particle cannot move. The wave exerts an upward force on the support, and by Newton's Third Law, the support exerts an equal and opposite downward force on the string. This reflects the pulse upside down (inverted), representing a phase change of 180° (π radians). At a free boundary, the end particle is free to move (e.g., a ring sliding frictionless on a rod). As the wave pulse arrives, the free end slides up, overshooting to twice the input amplitude due to inertia, and reflects the wave upright without inversion (0° phase shift).

Frequently Asked Questions

What is a transverse wave?
A transverse wave is a wave in which particles of the medium oscillate perpendicular (at right angles) to the direction of wave travel.
What are crests and troughs?
A crest is the point of maximum upward (positive) displacement from the equilibrium position. A trough is the point of maximum downward (negative) displacement.
How does tension affect wave speed on a string?
Wave speed is proportional to the square root of the tension (v ∝ √T). Increasing the tension of a string increases the speed of the transverse wave traveling along it.
What is Malus's Law?
Malus's Law states that the amplitude of a polarized wave passing through an analyzer is proportional to the cosine of the angle &theta; between their axes: A = A<sub>0</sub> cos&theta;, while the intensity is proportional to the cosine squared: I = I<sub>0</sub> cos&sup2;&theta;.
What is an example of an electromagnetic transverse wave?
All electromagnetic waves, including visible light, radio waves, microwaves, ultraviolet light, and X-rays, are transverse waves that propagate through oscillating electric and magnetic fields.