Stationary Waves in Sonometer

A sonometer (the name reflecting its purpose of making sound-related measurements) is a laboratory device built specifically to demonstrate, and let students experimentally verify, the precise relationship between the frequency of a transverse standing wave formed in a stretched string and three physical properties of that string: its tension, its vibrating length, and its mass per unit length. It consists of a long hollow resonance box carrying a single uniform, thin metallic wire -- which is why the instrument is also called a monochord -- with one end of the wire fixed to a hook and the other end passing over a pulley down to a hanging weight (the load), so that adding or removing weights directly changes the wire's tension. Two movable wooden knife-edge bridges, conventionally labelled P and Q, rest on top of the box and can be slid to any desired positions, thereby setting whatever length of wire is left free to vibrate between them. When the wire is set vibrating, it forms a transverse stationary wave with nodes forced to sit precisely at the two knife-edges P and Q (since the wire cannot move there) and a single antinode forming in the vibrating segment between them; if this vibrating length is $l$, it corresponds to exactly half a wavelength, so $l=\lambda/2$, giving $\lambda=2l$. Combining this with the general string wave-speed formula $v=\sqrt{T/\mu}$ (where $T$ is tension and $\mu$ is mass per unit length) yields the sonometer's central working formula, $f=\dfrac{1}{2l}\sqrt{\dfrac{T}{\mu}}$; expressing the mass per unit length in terms of the wire's material density $\rho$ and diameter $d$ via $\mu=\pi\rho d^2/4$ recasts this as $f=\dfrac{1}{ld}\sqrt{\dfrac{T}{\pi\rho}}$. In practice, the two knife-edges are slid until the vibrating segment resonates lou …