3.2.1.6. Amplitude

Table of Contents

Meaning of Amplitude

Amplitude tells us how large a wave’s disturbance is compared with its undisturbed position. In simple terms, it measures the maximum size of the vibration. If a rope is at rest along a straight line, then the amplitude of a wave on the rope is the greatest distance any point of the rope moves above or below that rest line.

For a sound wave, the idea is similar, but the disturbance is not usually sideways motion. Instead, the air experiences changes in pressure and density. In that case, amplitude describes the maximum pressure change, maximum density change, or maximum particle displacement from equilibrium, depending on how the wave is being described.

Amplitude is therefore not “how long” a wave is and not “how fast” it travels. It describes the strength or size of the oscillation.

Amplitude is the maximum displacement of the medium from its equilibrium position.
It is always measured from the equilibrium position to the extreme value, not from crest to trough.

Amplitude on a Wave Shape

For a transverse wave, such as a wave on a string, amplitude is easy to see on a sketch. The crest is the highest point, the trough is the lowest point, and the amplitude is the vertical distance from the equilibrium line to a crest or to a trough.

If the vertical distance from crest to trough is known, then the amplitude is half of that total height.

$$
A = \frac{\text{crest to trough height}}{2}
$$

Rendering scientific sketch…

Amplitude of a transverse wave

Mathematical Description

A wave is often written in a mathematical form such as

$$
y(x,t) = A \sin(kx - \omega t)
$$

Here, $A$ is the amplitude. It is the largest value that $y$ can reach in the positive or negative direction. Since the sine function varies between $-1$ and $1$, the displacement $y$ varies between $-A$ and $+A$.

So if

$$
y(x,t) = 0.03 \sin(kx - \omega t)
$$

then the amplitude is

$$
A = 0.03 \text{ m}
$$

which is 3 cm.

In an equation like $y(x,t) = A \sin(kx - \omega t)$ or $y(x,t) = A \cos(kx - \omega t)$, the amplitude is the coefficient in front of the sine or cosine.

Amplitude in Different Types of Waves

Amplitude appears in all waves, but what it means physically depends on the kind of wave.

Type of wave	What amplitude measures
Wave on a string	Maximum sideways displacement
Water surface wave	Maximum height change from equilibrium level
Sound wave	Maximum pressure, density, or particle displacement variation
Electromagnetic wave	Maximum value of electric field or magnetic field

This means amplitude is a general idea, not a single specific unit. Its unit depends on what quantity is oscillating. For displacement waves it may be meters. For sound pressure variation it may be pascals. For electric fields it may be newtons per coulomb or volts per meter.

Amplitude and Energy

A wave with larger amplitude carries more energy. This is one of the most important physical meanings of amplitude. If you shake a rope gently, the wave has small amplitude and carries less energy. If you shake it strongly, the amplitude is larger and the wave carries more energy.

In many cases, the energy carried by a wave is proportional to the square of the amplitude:

$$
E \propto A^2
$$

This means that if the amplitude doubles, the energy does not just double. It becomes four times as large.

Amplitude change	Energy change
$A \to 2A$	$E \to 4E$
$A \to 3A$	$E \to 9E$
$A \to \frac{1}{2}A$	$E \to \frac{1}{4}E$

For many waves, energy is proportional to the square of the amplitude:
$$
E \propto A^2
$$
A small increase in amplitude can mean a much larger increase in energy.

Amplitude and Intensity

Amplitude is closely related to intensity, which describes how much energy passes through an area in a given time. A larger amplitude usually means a greater intensity. For sound, this is connected with loudness. For light, it is connected with brightness.

Although the exact relationship depends on the type of wave, beginners should remember the main idea that stronger waves have larger amplitudes.

Common Mistakes

A common mistake is to confuse amplitude with wavelength. Wavelength measures the distance along the direction of travel between repeating points, while amplitude measures the size of the disturbance away from equilibrium.

Another common mistake is to measure amplitude from crest to trough. That full vertical distance is twice the amplitude, not the amplitude itself.

A third mistake is to think that amplitude changes the wave speed. In many basic situations, the wave speed depends on the medium, not on amplitude.

A Simple Example

Suppose a point on a string moves between $+0.04 \text{ m}$ and $-0.04 \text{ m}$ relative to equilibrium. Then the amplitude is

$$
A = 0.04 \text{ m}
$$

The total crest to trough distance is

$$
0.08 \text{ m}
$$

If another wave on the same string has amplitude $0.08 \text{ m}$, then its amplitude is twice as large. Its carried energy, in many simple models, is four times as large.

Visual Interpretation

Amplitude can be understood as the wave’s maximum “height” from its middle line. No matter where you look on the wave, the amplitude sets the limit of the disturbance. The wave may change with position and time, but the amplitude tells you the greatest possible size of that change.

Rendering scientific sketch…

Comparing small and large amplitude waves

Final Idea

Amplitude tells us how far the medium moves from equilibrium at most. It is a measure of wave size and is strongly connected to wave energy and intensity.

Key facts about amplitude:
Amplitude is the maximum disturbance from equilibrium.
It is measured from the equilibrium line to a crest or trough.
For a sinusoidal wave, it is the coefficient $A$ in the wave equation.
Larger amplitude means a more energetic wave.

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