Wave Mechanics: Partial Derivatives for Waves

A wave is a field that depends on more than one independent variable. In one spatial dimension we write

(1)

where x specifies position and t specifies time. Earlier articles used this notation geometrically and physically: a spatial snapshot is obtained by holding time fixed, while a time history is obtained by holding position fixed. WM12 also introduced the free-end condition

(2)

but deliberately postponed a systematic treatment of the derivative notation.

WM13 develops that calculus. The central idea is simple:

(3)

This is the mathematical language needed to describe local wave slope, local velocity, curvature, acceleration, and eventually the wave equation itself. Standard multivariable-calculus treatments emphasize that partial derivatives are ordinary rates of change taken one variable at a time, while standard wave texts interpret the resulting derivatives physically as slope, velocity, curvature, and acceleration [1, 2, 3, 4].

1 From one-variable derivatives to a field

For an ordinary function

(4)

there is only one independent variable. The derivative

(5)

asks how y changes when x changes.

For a wave field

(6)

there are two independent variables. We can ask two different local questions:

• How does u change from one nearby position to another at the same time?
• How does u change from one nearby time to another at the same position?

These questions lead to different derivatives.

2 The spatial partial derivative

At an event (x₀,t₀), hold the time fixed at t = t₀ and compare the field at two nearby positions:

(7)

As Δx approaches zero, this difference quotient approaches the spatial partial derivative:

(8)

The time t₀ does not change during this limiting process.

Geometrically,

(9)

If u is transverse displacement measured in meters and x is measured in meters, then

(10)

The slope is dimensionless in that particular application. For other wave variables, the units depend on the units of the field itself.

3 The temporal partial derivative

Now hold position fixed at x = x₀ and compare the field at two nearby times:

(11)

Taking the limit gives

(12)

The position x₀ does not change during this limiting process.

For a transverse string displacement,

(13)

If u is measured in meters,

(14)

This is not the same as the propagation speed of the wave. The wave may move through the medium with speed c while individual material points move up and down with velocity ∂u∕∂t.

4 Notation

Several notations are common:

(15)

and

(16)

The compact subscript notation is especially useful when expressions become long. For example,

(17)

In this series, both notations will be used. The fraction-like notation often makes the physical variable being held fixed easier to see, while the compact notation keeps wave equations readable.

5 Second partial derivatives

The first spatial derivative gives slope. Differentiate again with respect to position:

(18)

This measures how rapidly the slope changes with position. In one-dimensional wave problems it is a measure of local curvature. OpenStax uses this interpretation directly when developing the linear wave equation [2].

For a string displacement,

(19)

Likewise, differentiate the temporal derivative again:

(20)

For transverse displacement,

(21)

with units

(22)

This pair,

(23)

is the essential mathematical structure behind the one-dimensional wave equation.

6 Example: derivatives of a sinusoidal traveling wave

Consider the right-moving wave

(24)

Define the phase

(25)

Then

(26)

When differentiating with respect to x, time is held fixed. The chain rule gives

ux	=	(27)
	= −A sin 𝜃	(28)
	= −Ak sin 𝜃.	(29)

Differentiate once more:

uxx	= −Ak cos 𝜃	(30)
	= −Ak2 cos 𝜃.	(31)

Since

(32)

we obtain

(33)

Now differentiate with respect to time while holding x fixed:

ut	=	(34)
	= −A sin 𝜃	(35)
	= Aω sin 𝜃.	(36)

A second time derivative gives

utt	= Aω cos 𝜃	(37)
	= −Aω2 cos 𝜃.	(38)

Thus

(39)

The two second-derivative identities are therefore

(40)

7 A first glimpse of the wave equation

For the sinusoidal traveling wave,

(41)

and

(42)

Eliminate u between them:

(43)

WM08 established

(44)

Therefore

(45)

This is the one-dimensional linear wave equation.

At this point, however, we have only shown that a sinusoidal traveling wave has this derivative relationship. We have not yet derived the wave equation from the mechanics of a physical medium. That distinction matters. The later string-dynamics article will use Newton’s second law and Tension to show why a stretched string obeys this equation and why

(46)

The present article supplies the calculus needed for that derivation.

8 The result is not restricted to sinusoids

Return to the general right-moving disturbance from WM06:

(47)

Let

(48)

Then

(49)

The chain rule gives

ux	= F′(ξ)	(50)
	= F′(ξ),	(51)

while

ut	= F′(ξ)	(52)
	= −cF′(ξ).	(53)

Thus

(54)

for any sufficiently smooth right-moving disturbance of the form F(x − ct).

Differentiate again:

(55)

and

(56)

Therefore

(57)

The same second-order relationship also holds for a left-moving disturbance G(x + ct). This explains why the wave equation naturally accommodates both propagation directions. OpenStax makes the same connection between translating wave functions and the linear wave equation [2].

9 Partial derivative versus total derivative

The notation

(58)

means that x is held fixed while time changes. This is appropriate for a sensor mounted at one location or for a particular material element on an ideal string labeled by its equilibrium coordinate x.

But suppose instead that we follow a moving observation point x = x(t). Then the measured quantity is

(59)

Its ordinary time derivative requires the multivariable chain rule:

(60)

The first term is the local time change at fixed position. The second appears because the observer is moving through a spatially varying field.

This distinction becomes important in fluid mechanics, Electromagnetism, continuum mechanics, and transport theory. For the present Wave Mechanics series, the key lesson is simply that ∂∕∂t means fixed spatial coordinate, whereas d∕dt may describe a path through the (x,t) domain.

10 Mixed partial derivatives

A field can also be differentiated once with respect to each independent variable:

(61)

or

(62)

For sufficiently smooth functions, these are equal:

(63)

This result is often called equality of mixed partial derivatives or Clairaut’s theorem. Mixed derivatives are not needed for the basic one-dimensional wave equation, but they appear frequently in more advanced field theories and coordinate transformations. Standard multivariable-calculus treatments cover these derivative rules in detail [1].

11 Worked example 1: spatial and temporal derivatives

Consider

(64)

with x in meters and t in seconds.

The spatial derivative is

(65)

The units are dimensionless because displacement has units of meters and x has units of meters.

The temporal derivative is

(66)

At a given event, u_x tells us the local slope of the string while u_t tells us the local transverse velocity of that material point.

12 Worked example 2: second derivatives

For the same wave,

(67)

and

(68)

Therefore

(69)

The propagation speed is

(70)

so

(71)

Thus

(72)

13 Worked example 3: a non-sinusoidal pulse

Let

(73)

This has the translating form

(74)

with

(75)

Without carrying out the full algebra, the chain-rule result immediately gives

(76)

and

(77)

This is an important conceptual point: the derivative relationship is a property of translating waveforms, not only of sine and cosine functions.

14 Common mistakes

• Mistake: differentiating both x and t at once. A partial derivative changes one independent variable while holding the others fixed.
• Mistake: interpreting u_t as the propagation speed of the wave. For a string, u_t is the transverse velocity of a material point; the propagation speed is c.
• Mistake: treating u_x as a time derivative because the wave is moving. u_x is the slope of a spatial snapshot at fixed time.
• Mistake: forgetting the chain rule when differentiating cos(kx − ωt + ϕ).
• Mistake: missing the minus sign in u_xx = −k²u or u _tt = −ω²u.
• Mistake: assuming the appearance of u_tt = c²u _xx here is already a complete physical derivation of the string wave equation. WM13 verifies the differential relationship for translating waves; the later mechanics derivation explains why a real stretched string obeys it.

15 What WM13 adds to the wave-mechanics language

WM13 turns the graphical ideas of snapshot and time history into calculus.

At fixed time,

(78)

and

(79)

At fixed position,

(80)

and, for string displacement,

(81)

For a translating wave F(x − ct),

(82)

and

(83)

These tools prepare the way for the physical derivation of the one-dimensional string wave equation from force balance and Newton’s second law.

16 References

References