Wave Mechanics: Why Amplitude Is Not Energy

In elementary wave discussions one often hears statements such as

“The energy of a wave is proportional to the square of its amplitude.”

That statement is useful, but incomplete. It is a scaling statement, not an identification of amplitude with energy.

For a sinusoidal traveling wave on a particular string at a particular frequency, average energy density and average power do indeed scale as A². But amplitude by itself does not determine either quantity. Frequency, wave speed, linear density, Tension, and even the type of wave field can matter [1, 2, 4, 6].

The central message of this article is therefore

|------------------------| -Amplitude--is-not-energy.-

(1)

More precisely,

|----------------------------------------------------------------------------------------| |For a specified sinusoidal wave in a specified medium, energy and power often scale as A2.| ------------------------------------------------------------------------------------------

(2)

Those two statements are not contradictory. WM23 explains the distinction carefully.

1 Amplitude is a kinematic measure

For a transverse string wave u(x,t), the displacement amplitude A measures the largest magnitude of transverse displacement for a simple sinusoidal wave:

|u|max = A.

(3)

Its SI unit is length:

[A ] = m.

(4)

Energy, by contrast, has units

[E ] = J.

(5)

Energy per unit length has units

[ℰ] = J∕m,

(6)

and power has units

[P ] = W.

(7)

Dimensional analysis alone therefore shows that amplitude cannot literally be energy.

The more important physical distinction is that amplitude describes how far the field moves from equilibrium, while energy describes a combination of motion and deformation.

2 The string-energy formula contains derivatives, not displacement alone

WM18 derived the energy density of an ideal stretched string:

|------------------------| | 1- 2 1- 2 | -ℰ(x,-t)-=-2-μut-+-2-Tu-x.|

(8)

The first term is kinetic energy per unit length,

1 2 𝒦 = --μut, 2

(9)

and the second is elastic potential energy per unit length,

1 2 𝒰 = -T ux. 2

(10)

Notice what does not appear directly: the displacement u itself.

The energy depends on

how rapidly the string moves, through u_t,
how strongly it is locally tilted or deformed, through u_x,
the medium parameters μ and T.

This is the first major reason amplitude alone cannot determine energy.

3 Why the square of amplitude appears for a sinusoidal wave

Consider the right-moving sinusoidal wave

u(x,t) = A cos(kx − ωt + ϕ).

(11)

Define

𝜃 = kx − ωt + ϕ.

(12)

Then

ut = A ω sin 𝜃

(13)

and

u = − Ak sin𝜃. x

(14)

Substituting into the energy-density formula gives

ℰ

μA²ω² sin ²𝜃 + 1-
2

TA²k² sin ²𝜃.

(15)

For an ideal string,

T c2 = --, ω = ck, μ

(16)

T k2 = μω2.

(17)

Therefore

|-----------------| ℰ = μA2 ω2sin2𝜃. | -------------------

(18)

Averaging over a complete cycle gives

|---------------| | 1 2 2 | ⟨ℰ ⟩ = -μA ω . | -------2---------

(19)

WM20 similarly obtained

|----------------| |⟨P ⟩ = 1-μA2 ω2c.| -------2----------

(20)

The factor A² appears because both u_t and u_x are proportional to A, while kinetic and elastic energies are quadratic in those quantities.

This is the origin of the square law.

4 The phrase “proportional to amplitude squared” has hidden conditions

From

1 ⟨P ⟩ = --μA2 ω2c, 2

(21)

we may correctly say

|----------| |⟨P ⟩ ∝ A2 | -----------

(22)

only when the other relevant quantities are held fixed.

That means, for this formula, holding fixed

μ, ω, c.

(23)

Equivalently, if the same string and the same frequency are being compared, doubling amplitude multiplies average power by four.

But if frequency or the medium changes at the same time, amplitude alone is insufficient.

Figure. The amplitude A is only one input to the average-power relation. Frequency and medium properties are also required.

5 Counterexample 1: same amplitude, different frequency

Suppose two sinusoidal traveling waves exist on the same ideal string and have the same amplitude A. Let their frequencies be f and 2f.

Because

⟨P⟩ = 2π2 μA2f 2c,

(24)

the ratio of powers is

2 ⟨P2⟩-= (2f-)-= 4. ⟨P1⟩ f2

(25)

Thus

|--------------------------------------------| |equal amplitude does not imply equal power. | ----------------------------------------------

(26)

Figure. Two waves can have the same displacement amplitude while carrying different average power. On the same string, doubling frequency at fixed amplitude multiplies average power by four.

The physical reason is that the higher-frequency wave moves the string transversely more rapidly. Since kinetic energy depends on velocity squared, that increased motion matters strongly.

6 Counterexample 2: same amplitude and frequency, different medium

Now suppose two sinusoidal waves have the same amplitude and angular frequency but travel in different media.

The average power is

1 ⟨P ⟩ = --μA2 ω2c. 2

(27)

Using the characteristic impedance from WM22,

Z0 = μc,

(28)

we may write

|----------------| | 1- 2 2 | |⟨P ⟩ = 2 Z0A ω .| ------------------

(29)

Therefore two waves with identical A and ω can carry different power whenever their characteristic impedances differ.

Again,

|----------------------| |A alone is not enough. | ------------------------

(30)

7 Counterexample 3: zero displacement can coincide with maximum energy density

A particularly useful misconception check comes from the sinusoidal traveling wave itself.

For

u = A cos 𝜃,

(31)

the energy density is

2 2 2 ℰ = μA ω sin 𝜃.

(32)

π 𝜃 = -, 2

(33)

we have

u = 0,

(34)

but

sin2 𝜃 = 1.

(35)

Thus

|------------------------| u = 0 while ℰ = ℰmax.| --------------------------

(36)

At a crest or trough,

|u | = A,

(37)

but u_t = 0 and u_x = 0 for the traveling sinusoid at that instant, so

------------------------- | | -|u| =-A---while--ℰ-=-0.-|

(38)

Figure. For a sinusoidal traveling wave, the displacement itself and the local energy density do not peak at the same phase. Zero displacement can coincide with maximum local energy density.

This example makes the distinction especially clear: instantaneous displacement magnitude is not a local energy meter.

8 Standing waves make the distinction even sharper

Consider the Standing Wave

u(x,t) = 2A cos(kx) cos(ωt).

(39)

At nodes,

cos(kx ) = 0,

(40)

u = 0

(41)

for all time.

It might therefore be tempting to say that a node contains no wave energy. That conclusion is wrong.

The slope is

ux = − 2Ak sin (kx )cos(ωt).

(42)

At a node, | sin(kx)| = 1, so the elastic potential energy density becomes

1 2 2 2 2 𝒰 = -T ux = 2T A k cos (ωt). 2

(43)

This is generally nonzero and can reach a maximum.

Figure. A standing-wave node has zero displacement amplitude, yet it can store substantial elastic energy because the string slope is large there.

Thus even zero displacement amplitude at a point does not imply zero local energy density.

9 Superposition: cancellation of displacement is not disappearance of energy

Suppose a right-moving and a left-moving wave overlap:

u(x,t) = F (x − ct) + G (x + ct).

(44)

WM18 showed that the total energy density is

|---------′-------2-------′--------2-| -ℰ-=--T[F-(x-−-ct)]-+-T-[G-(x-+-ct)]-.|

(45)

The cross terms cancel between kinetic and elastic contributions.

Therefore the two component waves can cancel in displacement at some location and time,

F + G = 0,

(46)

without their total energy vanishing.

This is another reason not to infer energy directly from the instantaneous observed displacement.

10 Amplitude squared is a scaling factor, not an energy variable

The safest interpretation of statements such as

2 E ∝ A

(47)

is:

Within a specified wave model, if every relevant parameter except amplitude is held fixed, the energy measure under discussion changes as the square of amplitude.

For the ideal sinusoidal string wave,

1 ⟨ℰ ⟩ = -μA2 ω2 2

(48)

and

⟨P ⟩ = 1-μA2 ω2c. 2

(49)

Therefore

A → qA

(50)

implies

|------------------------------| ⟨ℰ ⟩ → q2⟨ℰ⟩, ⟨P⟩ → q2⟨P⟩,| --------------------------------

(51)

provided the other parameters remain unchanged.

This is the precise meaning of the square law.

11 What information is needed to infer energy from amplitude?

If the wave is known to be a sinusoidal traveling wave on an ideal string, the amplitude becomes useful only when accompanied by additional information.

For average energy density, one needs

A, ω, μ,

(52)

because

⟨ℰ ⟩ = 1μA2 ω2. 2

(53)

For average power, one additionally needs the wave speed, or equivalently the characteristic impedance:

1- 2 2 ⟨P ⟩ = 2μA ω c

(54)

⟨P ⟩ = 1-Z A2 ω2. 2 0

(55)

Thus an amplitude measurement can contribute to an energy estimate, but it does not by itself provide that estimate.

12 Worked Example 1: same amplitude, different frequency

Two right-moving sinusoidal waves travel on the same string. They have the same amplitude,

A = 2.0mm,

(56)

but their frequencies are

f1 = 20 Hz, f2 = 60 Hz.

(57)

Find the ratio of their average powers.

For the same string and the same amplitude,

⟨P ⟩ ∝ f 2.

(58)

Therefore

	= ²	(59)
	= ²	(60)
	= 9.	(61)

Hence

|--------------| |⟨P ⟩ = 9⟨P ⟩. | ---2--------1--

(62)

The waves have identical displacement amplitude, but one carries nine times the average power.

13 Worked Example 2: same amplitude and frequency, different string

Two right-moving sinusoidal waves have

A = 1.5 mm

(63)

and

f = 40Hz.

(64)

Wave 1 travels on a string with

μ1 = 0.010 kg/m, c1 = 100m/s,

(65)

while wave 2 travels on a string with

μ = 0.020 kg/m, c = 150m/s. 2 2

(66)

Because A and ω are the same,

⟨P2-⟩ μ2c2- ⟨P1 ⟩ = μ1c1.

(67)

Thus

	=	(68)
	= 3.	(69)

Therefore

|--------------| |⟨P2⟩ = 3⟨P1 ⟩. | ---------------

(70)

Equal amplitude and equal frequency still do not guarantee equal power when the medium changes.

14 Worked Example 3: zero displacement, maximum local energy density

A right-moving sinusoidal wave is

u = A cos(kx − ωt ).

(71)

At a particular point and time,

kx − ωt = π-. 2

(72)

The displacement is

( ) u = A cos π- = 0. 2

(73)

But

( ) ℰ = μA2 ω2 sin2 π- = μA2 ω2. 2

(74)

Hence

|------------------------| u = 0 while ℰ = ℰmax.| --------------------------

(75)

The zero displacement does not indicate zero energy.

15 Worked Example 4: a standing-wave node stores elastic energy

Consider

u(x,t) = 2A cos(kx) cos(ωt).

(76)

Choose a node where

π kx = --. 2

(77)

At the node,

u = 0

(78)

for all time.

However,

u = − 2Ak sin (kx )cos(ωt), x

(79)

so at the node,

ux = − 2Ak cos(ωt ).

(80)

The elastic energy density is

𝒰	= Tu_x²	(81)
	= T(4A²k² cos ²ωt)	(82)
	= 2TA²k² cos ²ωt.	(83)

Its maximum is therefore

----------------- | 2 2 | -𝒰max-=-2T-A--k-.|

(84)

The displacement amplitude at the node is zero, yet the local elastic energy can be large.

16 Worked Example 5: infer amplitude from a target average power

A sinusoidal traveling wave on a string has

μ = 0.012 kg/m, c = 90m/s, f = 30 Hz.

(85)

What amplitude is required to carry

⟨P ⟩ = 1.5 W?

(86)

Use

⟨P⟩ = 2π2 μA2f 2c.

(87)

Solve for A:

∘ --------- A = --⟨P-⟩--. 2π2μf 2c

(88)

Substituting,

∘ -------------------- 1.5 A = ---2----------2-----. 2π (0.012 )(30 )(90)

(89)

This gives approximately

|------------------------------| |A ≈ 8.84 × 10− 3m = 8.84mm. | --------------------------------

(90)

The example demonstrates the reverse point: amplitude can be inferred from energy transport only after frequency and medium properties are specified.

17 Worked Example 6: destructive displacement interference does not erase energy

Let

u (x,t) = F (x − ct) + G(x + ct)

(91)

with, at one particular point and time,

F = +a, G = − a.

(92)

The total displacement is

u = F + G = 0.

(93)

Suppose at that same event

′ ′ F = q, G = q.

(94)

The energy density is

′2 ′ 2 ℰ = T(F ) + T (G ).

(95)

Therefore

|--------2-| -ℰ-=-2T-q-.-

(96)

Although the displacement cancels exactly, the field can still contain nonzero energy.

18 Common mistakes

Mistake: saying “amplitude squared equals energy.” The dimensions are wrong, and the physical statement is incomplete.
Mistake: comparing two amplitudes without checking whether frequency and medium properties are the same.
Mistake: assuming u = 0 means local wave energy is zero.
Mistake: assuming a standing-wave node contains no energy because its displacement is always zero.
Mistake: treating destructive interference of displacement as destruction of conserved energy.
Mistake: using the sinusoidal square-law relation outside the assumptions under which it was derived.

19 Summary

Amplitude is a measure of field excursion. Energy is a mechanical quantity built from motion, deformation, and medium properties.

For the ideal string,

|------------------| | 1- 2 1- 2 | ℰ = 2μu t + 2T ux.| --------------------

(97)

For a sinusoidal traveling wave,

|--------------| |⟨ℰ ⟩ = 1μA2 ω2 | -------2--------

(98)

and

|----------------| | 1- 2 2 | |⟨P ⟩ = 2 μA ω c.| ------------------

(99)

Thus the statement

2 ⟨P ⟩ ∝ A

(100)

is correct only as a conditional scaling law when the other relevant quantities are fixed.

The broader lesson is

|-----------------------------------------------| Do--not-infer-wave-energy-from--amplitude-alone.--

(101)

This distinction becomes increasingly important in acoustics, electromagnetic waves, optics, and quantum wave mechanics, where different wave variables can have different relationships to transported or stored energy.

References

[1] A. P. French, Vibrations and Waves, M.I.T. Introductory Physics Series, W. W. Norton & Company, 1971.

[2] Frank S. Crawford, Jr., Waves, Berkeley Physics Course, Volume 3, McGraw-Hill, 1968.

[3] Howard Georgi, The Physics of Waves, Prentice Hall, 1993.

[4] William Moebs, Samuel J. Ling, and Jeff Sanny, University Physics, Volume 1, OpenStax, 2016, Section 16.4, “Energy and Power of a Wave.”

[5] Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Volume I, Chapters 47–49 on wave motion and modes.

[6] Massachusetts Institute of Technology, 8.03SC Physics III: Vibrations and Waves, MIT OpenCourseWare, materials on traveling waves, standing waves, and energy transport.

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