The Physics of Wave Speed

The wave velocity on a string is derived by applying Newton’s Second Law to a small string element Δl in a reference frame where the pulse is stationary. By equating the net vertical tension force F = T(Δl / R) to the centripetal force (μ Δl)(v²/R), the velocity is determined as v = √(T/μ).

The "Novel Insight": Unlike standard textbooks that often jump straight to the calculus of transverse displacement, this lecture uses the "Car Window" analogy. By sprinting alongside the pulse, we transform a complex moving wave into a stationary "hill" that the string material must flow over, making the use of centripetal acceleration (a = v²/R) intuitively obvious rather than mathematically forced.

"Imagine you are running right alongside the pulse... the pulse isn't moving at all—it’s just sitting there. But the string itself is now rushing past you".

"The frequency of the wave is fixed entirely by whatever generates the wave... velocity has no dependence on frequency".

Key Concepts

Transverse Tension: The restoring force F = 2T sin θ that drives the string back toward the equilibrium position.

Small Angle Approximation: The simplification sin θ ≈ θ (in radians), which allows for the geometric cancellation of the arc length Δl during the derivation.

Centripetal Force Balance: The requirement that the net inward tension provides the exact force needed for string particles to round the "curve" of the pulse at speed v.

Medium vs. Source: The distinction that tension T and linear density μ (properties of the medium) determine speed, while the generator (the source) determines frequency f.

Frequently Asked Questions (FAQ)

Q1. Why do we ignore gravity (mg) in the wave speed derivation?

In most physical strings, such as those on a guitar, the tension T is significantly larger than the weight mg of a tiny segment Δl. Including gravity would add negligible precision while making the vector geometry unnecessarily complex for finding the wave speed v.

Q2. Does the wave speed change if I pluck the string harder?

No. Plucking harder increases the amplitude and energy, but as long as the tension T and linear density μ remain constant, the speed v at which the pulse travels remains exactly the same.

Q3. If v = fλ, doesn't increasing the frequency increase the speed?

This is a common "circular logic" error. In a fixed medium, v is constant. If you increase the frequency f (by moving your hand more rapidly), the wavelength λ must decrease proportionally to keep v the same.