Standing Waves

A standing wave forms when two waves of the $\text{same frequency and amplitude}$ travel in $\text{opposite directions}$ and superpose — typically a wave and its reflection.

At some points the two waves always cancel: these are $\text{nodes}$, with zero displacement at all times. Midway between, they always reinforce: these are $\text{antinodes}$, with maximum amplitude.

Unlike a travelling wave, a standing wave transfers no net energy along the string. Energy is stored, oscillating between kinetic and potential forms.

A string fixed at both ends must have a node at each end. The simplest pattern — the $\text{fundamental}$ or first harmonic — is a single loop, so the string length is half a wavelength: $L=\lambda /2$.

The next pattern has two loops ($L=\lambda$), then three ($L=3\lambda /2$), and so on. In general the $n$th harmonic has $n$ loops and $L=n\lambda /2$.

Since the wave speed on the string is fixed, the frequencies are whole-number multiples of the fundamental.

The speed of a transverse wave on a string depends on the tension $T$ and the $\text{mass per unit length}$ $\mu$: $v=\sqrt{T/\mu }$.

Higher tension makes the string restore itself faster, raising the speed; a heavier string responds more sluggishly, lowering it.

Combining with the fundamental gives $f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$ — the three ways to tune a string: shorter, tighter or lighter all raise the pitch.

In the practical, a string passes over a pulley with masses hanging from it to set the tension, and a vibration generator drives one end. The frequency is adjusted until clear standing-wave patterns appear.

Varying the length or the tension and recording the resonant frequency tests the relationships above. A graph of $f$ against $\sqrt{T}$ should be a straight line through the origin with gradient $\frac{1}{2L\sqrt{\mu }}$.

Mass per unit length is found by weighing a measured length of the string. Tension is the weight of the hanging masses, $mg$.