Describe what is meant by wave motion as illustrated by vibrations in ropes and springs, and by experiments using water waves

3.1 General Properties of Waves

Objective

Describe what is meant by wave motion as illustrated by vibrations in ropes and springs, and by experiments using water waves (including reflection, refraction and diffraction).

1. Definition of a wave (syllabus wording)

A wave is a disturbance that transfers energy from the source to another part of the medium without the permanent transport of matter. The disturbance propagates through the medium by successive particle motions.

2. Key terminology

TermDefinition / Symbol
MediumThe material (solid, liquid or gas) through which a wave travels.
Amplitude (A)Maximum displacement of a particle from its equilibrium position.
Wavelength (λ)Distance between two successive points that are in the same phase (e.g., crest‑to‑crest or trough‑to‑trough).
Frequency (f)Number of complete cycles that pass a given point each second (Hz).
Period (T)Time for one complete cycle; \(T=\dfrac{1}{f}\).
Wave‑speed (v)Speed at which the disturbance travels; \(v=f\lambda=\dfrac{\lambda}{T}\).
Wave‑frontLine (or surface) joining points of equal phase. In a ripple tank the wave‑fronts are concentric circles.
CrestHighest point of a transverse wave (maximum positive displacement).
TroughLowest point of a transverse wave (maximum negative displacement).
PhaseThe position of a point within a wave cycle (e.g., 0°, 90°, 180°). Points of the same phase lie on the same wave‑front.
Direction of propagationThe direction in which the wave‑fronts move; it is perpendicular to the wave‑fronts.

3. Mandatory relationship

  \( \displaystyle v = f\lambda \)  (must‑know formula)

4. Transverse wave – vibrations in a rope

  • When the free end of a taut rope is moved up and down, a transverse wave travels away from the source; particles move perpendicular to the direction of propagation.

  • Wave‑speed depends on the tension \(T\) and the linear mass density \(\mu\):

    \[

    v = \sqrt{\frac{T}{\mu}}

    \]

  • Increasing the tension makes the wave travel faster; increasing \(\mu\) (e.g., using a thicker rope) makes it slower.

  • The driver’s frequency fixes the wave frequency; the wavelength adjusts so that \(v = f\lambda\) remains satisfied.

Rope with a transverse wave moving to the right

Rope fixed at the left, the right end oscillated by a hand. A crest moves to the right along the rope.

5. Longitudinal wave – vibrations in a spring

  • Compressing and releasing one end of a coiled spring produces a longitudinal wave; particles move parallel to the direction of propagation, creating alternating compressions and rarefactions.

  • Wave‑speed depends on the spring’s stiffness \(k\) and its linear mass density \(\mu\):

    \[

    v = \sqrt{\frac{k}{\mu}}

    \]

  • The driver’s frequency fixes the wave frequency; the spacing between successive compressions (the wavelength) changes so that \(v = f\lambda\) holds.

Spring with longitudinal wave

Series of compressed (dark) and stretched (light) sections moving away from a hand that oscillates the left end.

6. Water‑wave experiments (ripple tank)

6.1 Basic set‑up

  • A small paddle or vibrator is placed at the centre of a shallow tray of water (ripple tank).

  • The paddle is driven sinusoidally at a constant frequency \(f\).

6.2 Measuring wavelength and speed

  1. Wavelength (λ): Measure the distance between successive crests (or troughs) with a ruler placed on the water surface.

  2. Wave‑speed (v): Use a stopwatch to time how long a particular crest takes to travel a known distance \(d\); \(v = d/t\).

  3. Verify the mandatory relation \(v = f\lambda\).

6.3 Effect of water depth

  • Shallow water (depth \(h \ll \lambda\)): \(v = \sqrt{g\,h}\) where \(g = 9.81\;\text{m s}^{-2}\).

  • Deep water (depth \(\gg \lambda\)): \(v = \sqrt{\dfrac{g\lambda}{2\pi}}\).

6.4 Demonstrating the three characteristic phenomena

  • Reflection: Place a vertical barrier in the tank. Incident wave‑fronts strike the barrier and a reflected set of circular wave‑fronts emerges, obeying the law of reflection (angle of incidence = angle of reflection).

  • Refraction: Create a step in the bottom so that one side of the tank is shallower. As the wave passes from deep to shallow water its speed changes (according to \(v=\sqrt{gh}\)), causing the wave‑fronts to bend toward the normal.

  • Diffraction: Insert a narrow slit (width comparable to λ) in a barrier. Wave‑fronts emerging from the slit spread out into a semicircular pattern, illustrating that waves bend around obstacles comparable in size to their wavelength.

Top‑view of ripple tank showing incident, reflected, refracted and diffracted wave‑fronts

Top‑view diagram of a ripple tank showing (i) concentric incident wave‑fronts, (ii) reflected fronts from a barrier, (iii) bent fronts where depth changes (refraction), and (iv) spreading fronts after a narrow slit (diffraction).

7. Summary of wave‑motion principles

  • Wave motion transfers energy from the source to the medium without permanent displacement of the medium’s particles.

  • Key relationships (must‑know):

    • \( \displaystyle v = f\lambda \) (mandatory equation)

    • \( T = \dfrac{1}{f} \)

    • Transverse rope wave \( v = \sqrt{\dfrac{T}{\mu}} \)

    • Longitudinal spring wave \( v = \sqrt{\dfrac{k}{\mu}} \)

    • Shallow‑water surface wave \( v = \sqrt{g\,h} \)

    • Deep‑water surface wave \( v = \sqrt{\dfrac{g\lambda}{2\pi}} \)

  • Experiments with ropes, springs and water waves confirm how tension, mass per unit length, stiffness, frequency, depth and wavelength affect wave speed and wavelength.

  • Water‑wave experiments also demonstrate the three characteristic wave phenomena required by the syllabus:

    • Reflection – angle of incidence equals angle of reflection.

    • Refraction – change of direction when wave speed changes (e.g., at a depth step).

    • Diffraction – spreading of wave‑fronts when they encounter an opening comparable to λ.