# Wave Properties and the Wave Equation

Physics I · Principles of Waves, Fluids and Thermodynamics · https://tryals.app/learn/physics-i/wave-properties-and-the-wave-equation

Do the stadium wave: you stand up and sit down, yet the pattern races around the stadium. Nothing material travels, what moves is a **wave**: a disturbance that transfers energy without transferring matter.

**Types of waves**

| Type | Motion | Example |
| --- | --- | --- |
| **Transverse** | Displacement $\perp$ propagation | Light, strings |
| **Longitudinal** | Displacement $\parallel$ propagation | Sound |

**The wave equation** governing this motion is:

$$\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}$$

## Wave Parameters and Energy

A **travelling wave** is given by $y(x,t) = A\sin(kx - \omega t + \phi)$. Here, **Amplitude** ($A$) is max displacement, **wave number** ($k = 2\pi/\lambda$), and **angular frequency** ($\omega = 2\pi f$).

Speed is $v = f\lambda = \omega/k$. **Intensity** ($I$) is power per unit area:

$$I = \frac{1}{2}\rho v \omega^2 A^2$$

Intensity drops as $1/r^2$ for a point source (inverse-square law).

**Common pitfall:** Wave speed $v$ is a property of the **medium** (tension, density), not the source. Shaking a rope harder changes $A$, not $v$.

## Practice questions

6 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A wave has frequency **4 Hz** and wavelength **2 m**. Set the slider to its speed, using $v = f\lambda$.

**Answer:** 8 (within ±0.5)

**Why:** $v = 8$ m/s: crest spacing times crests-per-second is distance-per-second. This identity holds for every wave in nature — sound, light, ripples — because it is really just bookkeeping of crests.

Page: https://tryals.app/practice/physics-i/wave-properties-and-the-wave-equation/a-wave-has-frequency-4-hz-and-wavelength-2-m-set-the-slider-to-its

### 2. Match each wave quantity to its SI unit.

**Answer:**

- Frequency → Hz
- Wavelength → m
- Wave speed → m/s
- Period → s

**Why:** Frequency in hertz, wavelength and speed in SI distance/time units, period in seconds.

Page: https://tryals.app/practice/physics-i/wave-properties-and-the-wave-equation/match-each-wave-quantity-to-its-si-unit

### 3. Which of the following correctly describe transverse waves?

A. The medium oscillates parallel to the direction of propagation
B. Sound waves in air are transverse waves
C. The medium oscillates perpendicular to the direction of propagation
D. Electromagnetic waves in vacuum are transverse

**Answer:** C. The medium oscillates perpendicular to the direction of propagation; D. Electromagnetic waves in vacuum are transverse

**Why:** Transverse waves have displacement perpendicular to propagation; EM waves are transverse. Sound is longitudinal.

Page: https://tryals.app/practice/physics-i/wave-properties-and-the-wave-equation/which-of-the-following-correctly-describe-transverse-waves

### 4. Sort each wave into Transverse or Longitudinal.

**Answer:**

- Transverse: Radio waves, Seismic S-waves, Waves on a rope
- Longitudinal: Sound in water, Seismic P-waves, Compression pulse in a spring

**Why:** EM waves and shear waves are transverse; pressure/compression waves are longitudinal.

Page: https://tryals.app/practice/physics-i/wave-properties-and-the-wave-equation/sort-each-wave-into-transverse-or-longitudinal

### 5. A wave travels along a rope. You shake the end **twice as fast** (double the frequency) without changing the rope or its tension. What happens to the wavelength?

A. It doubles along with the frequency
B. It halves: the speed is fixed by the rope
C. It stays the same; only the speed doubles
D. It depends on the amplitude of the shaking

**Answer:** B. It halves: the speed is fixed by the rope

**Why:** The medium dictates $v$; the source dictates $f$. The wavelength is the negotiated result: $\lambda = v/f$. Doubling the frequency squeezes the same-speed wave into crests half as far apart, a rule that holds for sound, light and water alike.

Page: https://tryals.app/practice/physics-i/wave-properties-and-the-wave-equation/a-wave-travels-along-a-rope-you-shake-the-end-twice-as-fast-double

### 6. When a wave passes from one medium to another its frequency stays the same but its wavelength changes. Why is frequency the quantity that survives?

A. Because it is set by the source and cycles cannot be lost
B. Because wavelength is harder to measure and so appears to change
C. Because frequency is a property of the wave and wavelength of the medium
D. Because energy is conserved, which fixes the frequency

**Answer:** A. Because it is set by the source and cycles cannot be lost

**Why:** Crests cannot accumulate at an interface. With $v = f\lambda$ and $f$ fixed, a change in speed must appear in the wavelength, which is what refraction is.

Page: https://tryals.app/practice/physics-i/wave-properties-and-the-wave-equation/when-a-wave-passes-from-one-medium-to-another-its-frequency-stays-the