Formula of Amplitude: A = √(x² + v²/ω²), SHM and Waves

The formula of amplitude for simple harmonic motion is $A = \sqrt{x^2 + \left(\frac{v}{\omega}\right)^2}$ : from the displacement x and velocity v at any one instant, and the angular frequency ω, you can recover the largest displacement the oscillator ever reaches. Amplitude is measured in metres, and it is the number A that appears in front of the sine in $x = A\sin(\omega t + \phi)$ .

In words, amplitude is the maximum distance from the rest position. On a graph it is the height of a crest above the centre line, or half the distance from crest to trough. Several other routes lead to it: from energy, $A = \sqrt{\frac{2E}{k}}$ ; from maximum speed, $A = \frac{v_{\max}}{\omega}$ ; and straight from a wave equation. This page covers each amplitude formula in physics courses from Class 11 oscillations to AP Physics 1.

Quick facts

From x and v
$A = \sqrt{x^2 + \frac{v^2}{\omega^2}}$
From a graph
$A = \frac{y_{\max} - y_{\min}}{2}$
From energy
$A = \sqrt{\frac{2E}{k}}$
From maximum speed
$A = \frac{v_{\max}}{\omega}$
SI unit
metre (m) for a displacement amplitude
Dimensional formula
[M⁰ L¹ T⁰]
Energy link
Energy and intensity grow as A²

What is the formula of amplitude: the symbols

Symbols in the amplitude formula
Symbol	Quantity	SI unit
$A$	Amplitude, the largest displacement	m
$x$	Displacement at the chosen instant	m
$v$	Velocity at that same instant	m/s
$\omega$	How fast the phase advances, equal to $2\pi f$	rad/s
$E$	Total energy of the oscillator	J
$k$	Spring constant	N/m

The amplitude is always a positive size. A pendulum that swings 5 cm to the left and 5 cm to the right has an amplitude of 5 cm, not 10 cm. The full 10 cm is called the peak to peak value.

Where the formula comes from

In simple harmonic motion the velocity at displacement x is

$v = \omega\sqrt{A^2 - x^2}$

This comes from energy: at displacement x the energy $\frac{1}{2}kA^2$ splits into a moving part $\frac{1}{2}mv^2$ and a stored part $\frac{1}{2}kx^2$ , with $k = m\omega^2$ . Square both sides, divide by $\omega^2$ and rearrange:

$A^2 = x^2 + \frac{v^2}{\omega^2} \quad\Rightarrow\quad A = \sqrt{x^2 + \frac{v^2}{\omega^2}}$

Two checks: at the extreme position v = 0 and A = x, as it should. At the centre x = 0 and $A = \frac{v_{\max}}{\omega}$ , which is the maximum speed form.

Worked examples using the amplitude formula in physics

Example 1: from a single snapshot

A block on a spring oscillates with ω = 4.0 rad/s. At one instant it is 0.030 m from the centre, moving at 0.16 m/s. Find the amplitude.

$A = \sqrt{0.030^2 + \left(\frac{0.16}{4.0}\right)^2} = \sqrt{0.0009 + 0.0016} = \sqrt{0.0025} = 0.050\ \text{m}$

The block swings out to 5.0 cm on each side.

Example 2: from the equation of a wave

A wave on a rope is described by y = 0.04 sin(10πt − 2πx), with y in metres. What is its amplitude?

Compare with $y = A\sin(\omega t - kx)$ : A = 0.04 m = 4 cm. The other numbers give ω = 10π rad/s and the wave number 2π per metre, but they do not change the amplitude.

Example 3: using stored energy, answer in cm

A spring with k = 400 N/m stores 2.0 J of total energy in its oscillation. Find the amplitude in centimetres.

$A = \sqrt{\frac{2E}{k}} = \sqrt{\frac{2 \times 2.0}{400}} = \sqrt{0.010} = 0.10\ \text{m} = 10\ \text{cm}$

Example 4: from a graph

A displacement graph of a vibrating string shows crests at +6 mm and troughs at −6 mm. State the amplitude and the peak to peak value.

A = (6 − (−6)) ÷ 2 = 6 mm. Peak to peak = 12 mm.

Amplitude, energy and intensity

Energy in an oscillator is $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ . Twice the amplitude means four times the energy. The same square law holds for waves, which is why the intensity of wave formula contains $A^2$ . A sound wave with three times the amplitude is nine times as intense.

Amplitude does not affect the time period of an ideal spring or a pendulum at small angles, and it does not affect the speed of a wave. Those depend on the system and the medium. The frequency and amplitude of a wave are independent: you can change either one without changing the other.

Special cases

Two SHMs in the same direction. Combining amplitudes A₁ and A₂ with a phase difference φ gives $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi}$ . In phase, they add; exactly out of phase, they subtract.
Damped oscillations. Friction makes the amplitude shrink with time, $A(t) = A_0 e^{-bt/2m}$ , where b is the damping constant.
Standing waves. The amplitude changes from place to place: zero at nodes and largest at antinodes.
Resonance. When a driving force matches the natural frequency, the amplitude becomes very large.

Mistakes to avoid

Giving the peak to peak value as the amplitude. Halve it.
Mixing ω and f. In $A = \frac{v_{\max}}{\omega}$ , ω must be in rad/s; multiply f by 2π first.
Leaving centimetres in an energy formula. Convert to metres before you use $\frac{1}{2}kA^2$ .
Thinking a bigger amplitude means a faster wave. It means more energy, not more speed.

Spring energy is covered in depth on the spring potential energy page, and every other wave formula is one click away on the waves and optics hub.

Frequently asked questions

How do you calculate amplitude in SHM?

For SHM, A = √(x² + v²/ω²). You can also use A = vₘₐₓ / ω, A = √(2E/k), or read A directly from x = A sin(ωt + φ).

What is the SI unit of amplitude?

For a displacement wave or oscillation, the metre. For other waves the unit follows the quantity that oscillates, for example volts for an alternating voltage or pascals for a pressure wave.

How do you find amplitude from a graph?

Measure from the centre line to a crest. Equivalently, take the difference between the highest and lowest values and divide by 2.

Does amplitude affect frequency?

No, not for an ideal oscillator or wave. Frequency is set by the source or the system, and amplitude only sets how much energy is involved.

How are amplitude and energy related?

Energy is proportional to amplitude squared, E = ½kA². Doubling the amplitude gives four times the energy.