# Wave equations, normal modes and boundary conditions

> The ideal one-dimensional wave equation is ytt=c²yxx. The parameter c is propagation speed for this nondispersive model.

Course: [University mechanics and waves](https://studybrix.com/en/courses/university-mechanics/index.md)
Level: university
Prerequisites: Partial derivatives and trigonometric functions.

## Notes

### Wave equation

The ideal one-dimensional wave equation is ytt=c²yxx. The parameter c is propagation speed for this nondispersive model.

### Boundary data

Fixed endpoints require zero displacement at each end. They permit standing waves whose spatial wave numbers are nπ/L.

### Superposition

Solutions of a linear homogeneous wave equation can be added. The result need not resemble one single travelling sinusoid.

### Energy

A standing wave can have stationary nodes while its material elements oscillate. Node locations should not be confused with material travelling at wave speed.

## Practice questions

### 1. A fixed-fixed string has L=2 m and wave speed 20 m/s. Find its fundamental frequency.

Answer: 5 Hz

Explanation: f1=c/(2L)=20/4.

### 2. For the same ideal fixed-fixed model with f1=5 Hz, find f3.

Answer: 15 Hz

Explanation: fn=nf1; the third harmonic is 15 Hz.

### 3. What does a fixed endpoint impose on transverse displacement?

A. It is zero at that endpoint
B. The tension is zero
C. Its derivative in time is always one
D. It is maximal at all times

Answer: It is zero at that endpoint

Explanation: The boundary condition fixes displacement, not the whole field.

### 4. Which property justifies adding two wave solutions?

A. Identical initial phases
B. A moving boundary in every case
C. Large amplitude
D. Linearity of the governing homogeneous equation

Answer: Linearity of the governing homogeneous equation

Explanation: The linear operator applied to a sum equals the sum of its actions.

### 5. A wave has angular frequency 12 rad/s and wave number 3 rad/m. Find phase speed.

Answer: 4 m/s

Explanation: For a sinusoid, vp=ω/k.

### 6. Explain why normal-mode frequencies depend on boundary conditions as well as the wave equation.

Sample answer / marking guidance: The equation determines possible local motion, while boundary conditions select permitted spatial patterns and wave numbers. Different endpoint constraints select different spectra.

Explanation: Credit both the field equation and allowed-mode selection.

### 7. At a node of an ideal standing transverse wave, what happens to displacement?

A. It equals wave speed
B. It is always a maximum
C. It grows forever
D. It remains zero

Answer: It remains zero

Explanation: A node is a position with zero oscillation amplitude.

### 8. State why an ideal string model may fail at very large displacements.

Sample answer / marking guidance: The small-slope approximation and approximately constant tension can fail, introducing nonlinear terms and changing wave speed and mode behaviour.

Explanation: Credit explicit model assumptions, not merely saying all equations fail.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.