How can we derive the wave speed formula for a rope wave by comparing coefficients of the wave equation?

Conditions

• The wave can be written in the traveling-wave form $f(x-vt,0)$.
• The physical system is a string with tension $F_T$ and linear density $\mu$.

Reasoning, step by step

1. Compute the second-order partial derivatives of the traveling wave $f(x-vt,0)$ with respect to $x$ and $t$ using the chain rule.
2. Establish the mathematical relationship $\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$.
3. Recall the physical wave equation derived from force analysis: $\frac{\partial^2 f}{\partial x^2} = \frac{\mu}{F_T} \frac{\partial^2 f}{\partial t^2}$.
4. Compare the coefficients of the second time derivative in both equations.
5. Solve the resulting algebraic equation $\frac{1}{v^2} = \frac{\mu}{F_T}$ for $v$.

Example

The center of the screen highlights two second-order partial derivative expressions in a red box, with blue arrows pointing downward to the equation $\frac{1}{v^2} = \frac{\mu}{F_T} \implies v = \sqrt{\frac{F_T}{\mu}}$.

Common misconceptions

• Assuming the wave speed depends on the amplitude of the wave.
• Forgetting that the coefficient comparison relies on both equations describing the exact same physical phenomenon.

Watch the explanation

Explore next

• How is the chain rule applied when computing the second-order partial derivative of the traveling wave function f(x-vt) with respect to time t?
• What is the dimensional verification of the wave speed formula?