Stationary Waves in a Stretched String: MDCAT Physics notes
Stationary Waves in a Stretched String notes for MDCAT: harmonics, λ = 2L/n, f = nv/2L, v = √(F/m), counting nodes and loops, and common MDCAT traps.
Setting up the waves
A string of length $L$ is fixed at both ends and stretched with tension $F$. When it is plucked or vibrated, waves travel along it, reflect at the fixed ends, and the incident and reflected waves superpose to form a stationary wave. Both fixed ends must be nodes.
Speed of waves on a string
$$v = \sqrt{\frac{F}{m}} \quad\text{or}\quad v^2 = \frac{F}{m}$$
where $m$ is the mass per unit length ($\text{kg m}^{-1}$). A tighter or lighter string carries waves faster.
Modes of vibration (harmonics)
Since the string must contain a whole number $n$ of loops, each $\lambda/2$ long:
$$L = n\frac{\lambda_n}{2} \quad\Rightarrow\quad \lambda_n = \frac{2L}{n}$$
$$f_n = \frac{v}{\lambda_n} = \frac{nv}{2L} = n f_1, \qquad f_1 = \frac{1}{2L}\sqrt{\frac{F}{m}}$$
| Mode | Loops | Nodes | Antinodes | $\lambda$ | $f$ |
|---|---|---|---|---|---|
| Fundamental (1st harmonic) | 1 | 2 | 1 | $2L$ | $f_1$ |
| 2nd harmonic | 2 | 3 | 2 | $L$ | $2f_1$ |
| 3rd harmonic | 3 | 4 | 3 | $2L/3$ | $3f_1$ |
| nth harmonic | n | n + 1 | n | $2L/n$ | $nf_1$ |
So a string fixed at both ends gives all harmonics (odd and even). The number of loops $n$ equals the number of antinodes; nodes are one more. The total (nodes + antinodes) $= 2n + 1$ is always odd.
The harmonic number is the number of loops, not the number of nodes. Frequency $= (\text{number of loops}) \times f_1$ and wavelength $= 2L/(\text{number of loops})$.
Shape of the string
At any instant the string's shape is symmetric about its midpoint: for odd harmonics it is a mirror image; for even harmonics the two halves are equal and opposite (the midpoint is a node).
Worked examples
- A 2 m string vibrating in 2 loops: each loop $= 1\ \text{m} = \lambda/2$. Distance between consecutive nodes is 1 m and $\lambda = 2\ \text{m}$.
- Fundamental of 100 Hz: the 3rd harmonic is 300 Hz, showing 3 loops and 4 nodes.
- Quadrupling the tension doubles $v$ and so doubles every frequency.
- A string of length 0.5 m carries waves at $200\ \text{m s}^{-1}$: $f_1 = 200/(2 \times 0.5) = 200\ \text{Hz}$; halving the length would double this to 400 Hz.
Key formulas
- $v = \sqrt{F/m}$
- $\lambda_n = 2L/n$
- $f_n = \dfrac{n}{2L}\sqrt{F/m} = nf_1$
Common MDCAT traps
- Distance between consecutive nodes is $\lambda/2$, not one wavelength.
- $v^2 = F/m$, not $v = F/m$; $m$ is mass per unit length.
- Use the number of loops (not nodes) in $\lambda = 2L/n$ and $f = nf_1$.
- Nodes + antinodes is always odd for a string fixed at both ends.
Quick revision
- Fixed ends are nodes.
- Fundamental: $\lambda = 2L$.
- All harmonics are present on a stretched string.
- Nodes = loops + 1.
- Frequency ∝ $\sqrt{F}$ and ∝ $1/L$.