# Superposition, Interference, and Standing Waves

Physics I · Principles of Waves, Fluids and Thermodynamics · https://tryals.app/learn/physics-i/superposition-interference-and-standing-waves

## Superposition and Interference

Two waves can occupy the same place at the same time, adding point by point before walking away unchanged. The **superposition principle** states that the total displacement is the vector sum: $y_{\text{total}} = y_1 + y_2$.

When waves meet, **interference** occurs based on their phase difference $\Delta\phi$:

| Type | Condition | Result ||
| --- | --- | --- |
| **Constructive** | $\Delta\phi = 0, 2\pi, \ldots$ | Larger amplitude |
| **Destructive** | $\Delta\phi = \pi, 3\pi, \ldots$ | Cancellation |

**Beats** occur when two waves of slightly different frequencies combine, producing a modulation frequency of $f_{\text{beat}} = |f_1 - f_2|$.

> **Common pitfall:** At perfect destructive overlap, the string looks completely flat, but energy has not vanished. It hides in the string's **velocity**. Displacement cancels; energy does not.

## Standing Waves and Boundaries

Confining identical waves travelling in opposite directions forms a **standing wave**: $y(x,t) = 2A\sin(kx)\cos(\omega t)$. The pattern features fixed **nodes** (zero displacement) and **antinodes** (maximum displacement).

Standing waves only form at specific frequencies called **natural modes**. For a string of length $L$, boundary conditions dictate the allowed wavelengths $\lambda_n$ and resonant frequencies $f_n$:

| End Condition | Wavelength $\lambda_n$ | Frequencies $f_n$ |
| --- | --- | --- |
| **Both Ends Fixed** | $2L/n$ | $\frac{nv}{2L}$ |
| **One End Open** | $4L/n$ (odd $n$) | $\frac{nv}{4L}$ |

Resonant frequency depends on string properties: $f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$, where $T$ is **tension** and $\mu$ is **mass per unit length**.

## Practice questions

5 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort each standing-wave feature into Node or Antinode.

**Answer:**

- Node: Point of zero displacement, Fixed end of a string, Point of destructive superposition
- Antinode: Point of maximum displacement, Open end of an organ pipe (pressure), Centre of fundamental mode on a fixed string

**Why:** Nodes are zero-amplitude points; antinodes are maximum-amplitude points in a standing wave.

Page: https://tryals.app/practice/physics-i/superposition-interference-and-standing-waves/sort-each-standing-wave-feature-into-node-or-antinode

### 2. Two identical wave pulses — one upright, one inverted — travel toward each other on a string. At the instant they perfectly overlap, the string is **completely flat**. Where did the energy go?

A. Into the string's motion: flat but moving
B. It reflected back toward both sources
C. It cancelled permanently, like the displacements did
D. It converted to heat at the overlap point

**Answer:** A. Into the string's motion: flat but moving

**Why:** At perfect cancellation, every point has zero displacement but maximum speed, the pulses’ velocity fields reinforce. All the energy is momentarily kinetic, which is why the pulses re-emerge unchanged and keep going. Superposition erases nothing.

Page: https://tryals.app/practice/physics-i/superposition-interference-and-standing-waves/two-identical-wave-pulses-one-upright-one-inverted-travel-toward

### 3. A standing wave has nodes that never move and antinodes that swing hardest. Why does a travelling wave have neither?

A. Because the two have different frequencies
B. Because a standing wave is two waves travelling opposite ways
C. Because standing waves carry no energy and travelling waves do
D. Because standing waves occur only in bounded media

**Answer:** B. Because a standing wave is two waves travelling opposite ways

**Why:** Permanent stillness requires permanent cancellation, and that requires two waves. The reflected wave supplies the partner, which is why standing waves form on bounded strings and in pipes.

Page: https://tryals.app/practice/physics-i/superposition-interference-and-standing-waves/a-standing-wave-has-nodes-that-never-move-and-antinodes-that-swing

### 4. Order these milestones in the history of wave physics.

**Answer:**

1. Huygens proposes wave theory of light
2. Young demonstrates double-slit interference
3. Fresnel develops full wave optics (diffraction)
4. Maxwell unifies electricity, magnetism, and light as EM waves
5. Hertz experimentally confirms EM waves

**Why:** Wave theory developed from Huygens (1678) through Young (1801), Fresnel (1820s), Maxwell (1865) to Hertz (1887).

Page: https://tryals.app/practice/physics-i/superposition-interference-and-standing-waves/order-these-milestones-in-the-history-of-wave-physics

### 5. A guitar string 0.65 m long vibrates in its **fundamental** mode with wave speed 260 m/s (fundamental: $\lambda = 2L$). What is its frequency, in Hz? **Set the slider to your answer.**

**Answer:** 200 (within ±6)

**Why:** $f = 260/1.3 = 200$ Hz. Every fretted note is this same computation: shortening $L$ (fretting) or raising $v$ (tuning tighter) raises the pitch.

Page: https://tryals.app/practice/physics-i/superposition-interference-and-standing-waves/a-guitar-string-0-65-m-long-vibrates-in-its-fundamental-mode-with