Stationary wave
Waves - OCR A-Level Physics
Key Definition
Stationary wave
A wave pattern formed by the superposition of two progressive waves of the same frequency, amplitude, and type travelling in opposite directions. No energy is transferred along the wave.
A wave pattern formed by the superposition of two progressive waves of the same frequency, amplitude, and type travelling in opposite directions. No energy is transferred along the wave.
Key Definition
Node
A point on a stationary wave where the displacement is always zero. Formed by persistent destructive interference.
A point on a stationary wave where the displacement is always zero. Formed by persistent destructive interference.
Key Definition
Antinode
A point on a stationary wave where the displacement varies between maximum positive and maximum negative values. Formed by persistent constructive interference.
A point on a stationary wave where the displacement varies between maximum positive and maximum negative values. Formed by persistent constructive interference.
- Nodes and antinodes alternate, equally spaced along the wave.
- The distance between adjacent nodes (or adjacent antinodes) = λ/2.
- All points between two adjacent nodes oscillate in phase with each other.
- Points on opposite sides of a nodeA point on a stationary wave where the displacement is always zero. Located at half-wavelength intervals. are in antiphase (180° out of phase).
- Unlike progressive waves, stationary waves do NOT transfer energyThe capacity to do work. Measured in joules (J). - energyThe capacity to do work. Measured in joules (J). is stored between nodes.
Harmonics on a Stretched String
- Both ends of a stretched string are fixed, so both ends must be nodes.
- The first harmonicThe lowest frequency standing wave on a string or in a pipe, with the fewest nodes and antinodes. Also called the fundamental. (fundamental, $n = 1)$: one loop. $L = \lambda/2$, so $\lambda = 2L$.
- The second harmonicThe standing wave mode with twice the frequency of the first harmonic, having one additional node. (n = 2): two loops. $L = \lambda$, so $\lambda = L$.
- The third harmonicThe standing wave mode with three times the frequency of the first harmonic. (n = 3): three loops. $L = 3\lambda/2$, so $\lambda = 2L/3$.
- General pattern: L = nλ/2, giving $\lambda_{n} = 2L/n$ and $f_{n} = nf_{1}$.
Stationary Waves in Pipes
- Closed $end = nodeA point on a stationary wave where the displacement is always zero. Located at half-wavelength intervals. (air cannot move). Open end = antinodeA point on a stationary wave where the displacement is a maximum. Located midway between adjacent nodes.$.
- Open pipe (both ends open): all harmonics present. $f_{n} = nv/(2L)$ for n = 1, 2, 3, ...
- Closed pipe (one end closed): only odd harmonics. $f_{n} = nv/(4L)$ for n = 1, 3, 5, ...
- The fundamental of a closed pipe has $L = \lambda/4$.
Worked Example [3 marks]
A guitar string of length 0.64 m vibrates in its first harmonic with a frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). of 330 Hz. Calculate the wave speedThe distance travelled by a wavefront per unit time. on the string.
Show Solution
1
First harmonic
$L = \lambda/2, so \lambda = 2L = 2 \times 0.64 = 1.28\;\text{m}$
[1]2
$v = f\lambda = 330 \times 1.28$
[1]3
$v = 422\;\text{m} s⁻¹$
[1]Answer
$v = 422\;\text{m} s⁻¹$
Common Mistake
MEDIUM
Students often: Confusing the harmonic number with the number of nodes. Writing that the first harmonic has 1 nodeA point on a stationary wave where the displacement is always zero. Located at half-wavelength intervals..
Instead: The harmonic number n equals the number of loops (antinodes), NOT the number of nodes. A string at its first harmonic has 2 nodes (both ends) and 1 antinodeA point on a stationary wave where the displacement is a maximum. Located midway between adjacent nodes.. Number of $nodes = n + 1$ for a string fixed at both ends.
Instead: The harmonic number n equals the number of loops (antinodes), NOT the number of nodes. A string at its first harmonic has 2 nodes (both ends) and 1 antinodeA point on a stationary wave where the displacement is a maximum. Located midway between adjacent nodes.. Number of $nodes = n + 1$ for a string fixed at both ends.
Examiner Tips and Tricks
- When drawing stationary waveA wave pattern formed by the superposition of two progressive waves of the same frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). and amplitudeThe maximum displacement of a point on a wave from its equilibrium (rest) position. Measured in metres (m). travelling in opposite directions. EnergyThe capacity to do work. Measured in joules (J). is not transferred along a stationary wave. diagrams, always label nodes (N) and antinodes (A).
- Show the wave at two instants - maximum displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). and the mirror image - to demonstrate that it oscillates rather than travels.