Wave Mechanics Examples: Partial Derivatives for Waves

This companion article provides exercises for WM13, wave mechanics: Partial Derivatives for Waves. All exercises are presented first. Complete worked solutions follow in Part II.

WM13 introduced the calculus of a one-dimensional wave field

(1)

The first partial derivatives answer two different local questions:

(2)

and

(3)

For transverse string displacement, u_t is the transverse velocity of the material point labeled by x, while

(4)

measures spatial curvature and

(5)

is local transverse acceleration [1, 2, 3, 4].

For a right-moving translating disturbance

(6)

WM13 derived

(7)

and

(8)

The exercises below reinforce those results without treating this as the physical Newton’s-law derivation of the string wave equation. That derivation comes later in the series.

How to use this problem set

Attempt all exercises in Part I before consulting Part II. For each derivative, state explicitly which independent variable is being varied and which is being held fixed. When a derivative has physical units, include them. In chain-rule problems, identify the intermediate phase or translating coordinate before differentiating.

Part I: Exercises

Exercise 1: What is being held fixed?

For a field u(x,t), describe in words what each derivative means and state which independent variable is held fixed.

1. u_x
2. u_t
3. u_xx
4. u_tt

For transverse displacement of a string, give the physical interpretation of each derivative where appropriate.

Exercise 2: Units of wave derivatives

A transverse string displacement u is measured in meters, position x is measured in meters, and time t is measured in seconds.

Determine the units of

1. u_x,
2. u_t,
3. u_xx,
4. u_tt,
5. c²u _xx if c has units of meters per second.

Explain why the units in part (e) are consistent with the relation u_tt = c²u _xx.

Exercise 3: Estimate derivatives from finite differences

At time t₀, measurements near x₀ give

(9)

and

(10)

At the fixed position x₀, measurements at nearby times give

(11)

Use forward difference quotients to estimate

1. u_x(x₀,t₀),
2. u_t(x₀,t₀).

State the sign and physical meaning of each result.

Exercise 4: Read derivative signs from a spatial profile

The following spatial snapshot marks four points on a sinusoidal profile.

Without calculating exact numerical values, determine the sign of u_x and the sign of u_xx at points P, Q, R, and S.

Use the local slope for u_x and the local concavity for u_xx.

Exercise 5: Differentiate a sinusoidal traveling wave

Consider

(12)

Starting from the phase

(13)

derive expressions for

1. u_x,
2. u_xx,
3. u_t,
4. u_tt.

Then show that

(14)

and

(15)

Exercise 6: Numerical derivatives at one event

A traveling wave is

(16)

where x is in meters and t is in seconds.

At the event

(17)

find

1. u,
2. u_x,
3. u_t,
4. u_xx,
5. u_tt.

Interpret the signs of u_x and u_t physically.

Exercise 7: Verify the wave-equation relation for a sinusoid

For the wave in Exercise 6,

1. determine the propagation speed from c = ω∕k,
2. compute c²u _xx symbolically,
3. show that c²u _xx = u_tt,
4. explain why this verification is not yet the physical derivation of the string wave equation.

Exercise 8: Chain rule for an arbitrary right-moving profile

Let

(18)

Define

(19)

Use the chain rule to derive

1. u_x,
2. u_t,
3. u_xx,
4. u_tt.

Then prove

(20)

and

(21)

Exercise 9: Left-moving profile and the sign change

Now let

(22)

1. Derive the relationship between u_t and u_x.
2. Derive the relationship between u_tt and u_xx.
3. Explain why the first-order relation changes sign but the second-order wave-equation relation does not.

Exercise 10: Interpret the right-moving slope-rate relation

The following figure compares the spatial slope of a translating profile with its local time rate.

Answer the following.

1. If u_x > 0 at some event and c > 0, what is the sign of u_t?
2. If u_x < 0, what is the sign of u_t?
3. Give a geometric explanation using a right-moving pulse.
4. What relation would replace this one for a left-moving profile?

Exercise 11: Partial derivative versus total derivative

The figure below shows three directions through the (x,t) domain.

Suppose an observer moves according to x = x(t) through a field u(x,t).

1. State the multivariable chain-rule formula for du∕dt.
2. For a right-moving profile u = F(x−ct), substitute u_t = −cu_x into the total derivative.
3. What value of dx∕dt makes du∕dt = 0?
4. Interpret that result physically.

Exercise 12: Mixed partial derivatives

Consider

(23)

Calculate

1. u_x,
2. u_t,
3. u_xt,
4. u_tx.

Verify that u_xt = u_tx for this smooth function.

Exercise 13: Diagnose conceptual statements

For each statement, decide whether it is correct. If incorrect, rewrite it accurately.

1. “u_t is the speed at which the wave pattern propagates.”
2. “When calculating u_x, time must be held fixed.”
3. “If u_xx = 0 at a point, the displacement must also be zero there.”
4. “For a right-moving profile F(x − ct), u_t and u_x always have the same sign.”
5. “Showing that a trial waveform satisfies u_tt = c²u _xx proves that every physical medium obeys that wave equation.”

Exercise 14: Synthesis problem

A right-moving sinusoidal wave is

(24)

At the event

(25)

complete the following.

1. Identify A, k, ω, and ϕ.
2. Find λ, T, f, and c.
3. Find u_x and u_t at the event.
4. Verify numerically that u_t = −cu_x at that event.
5. Find u_xx and u_tt at the event.
6. Verify numerically that u_tt = c²u _xx.
7. In one sentence, distinguish the local material velocity u_t from the propagation speed c.

Part II: Complete Worked Solutions

Solution 1: What is being held fixed?

1. u_x is the spatial rate of change of the field. Position varies while time is held fixed. For string displacement it is the local slope of a spatial snapshot.
2. u_t is the local time rate of change. Time varies while position is held fixed. For string displacement it is the transverse velocity of the material point labeled by x.
3. u_xx is the rate at which spatial slope changes with position while time remains fixed. In one-dimensional wave problems it measures local curvature.
4. u_tt is the rate at which u_t changes with time while position remains fixed. For string displacement it is local transverse acceleration.

Solution 2: Units of wave derivatives

The field u has units of meters.

1. (26)

   For displacement versus position, the slope is dimensionless.

2. (27)

3. (28)

4. (29)

5. Since

   (30)

   we obtain

   (31)

   This is exactly the unit of u_tt, so the relation u_tt = c²u _xx is dimensionally consistent.

Solution 3: Estimate derivatives from finite differences

For the spatial derivative,

ux(x0,t0)	≈	(32)
	=	(33)
	= .	(34)

Converting 0.6 mm = 6.0 × 10⁻⁴ m,

(35)

The negative sign means the profile slopes downward as x increases at that instant.

For the time derivative,

ut(x0,t0)	≈	(36)
	=	(37)
	=	(38)
	= 0.080 m/s .	(39)

The positive sign means the local string point is moving in the positive transverse direction at that event.

Solution 4: Read derivative signs from a spatial profile

For a cosine-shaped spatial snapshot:

1. At P, the curve is descending, so u_x < 0. The profile is concave downward there, so u_xx < 0.
2. At Q, the profile crosses zero while descending most steeply, so u_x < 0. At the zero crossing of a pure cosine, the curvature is zero, so u_xx = 0.
3. At R, the profile is at a trough. The slope is zero, so u_x = 0, while the curve is concave upward, so u_xx > 0.
4. At S, the profile is rising, so u_x > 0. Since the displacement there is positive for a cosine and u_xx = −k²u, the curvature is negative: u _xx < 0.

Solution 5: Differentiate a sinusoidal traveling wave

Let

(40)

For the spatial derivative, hold t fixed:

ux	= (A cos 𝜃)	(41)
	= −A sin 𝜃	(42)
	= −Ak sin 𝜃.	(43)

Differentiate again:

uxx	= −Ak cos 𝜃	(44)
	= −Ak2 cos 𝜃	(45)
	= −k2u.	(46)

Thus

(47)

For the time derivative, hold x fixed:

ut	= (A cos 𝜃)	(48)
	= −A sin 𝜃	(49)
	= Aω sin 𝜃.	(50)

Differentiate again:

utt	= Aω cos 𝜃	(51)
	= −Aω2 cos 𝜃	(52)
	= −ω2u.	(53)

Therefore

(54)

Solution 6: Numerical derivatives at one event

The wave is

(55)

At

(56)

the phase is

𝜃	= 5(0.20) − 20(0.050)	(57)
	= 1.0 − 1.0	(58)
	= 0.	(59)

Therefore

(60)

1. (61)

2. (62)

   so

   (63)

3. (64)

   so

   (65)

4. uxx	= −k2u	(66)
   	= −25(0.030)	(67)
   	= −0.750 m−1 .	(68)

5. utt	= −ω2u	(69)
   	= −400(0.030)	(70)
   	= −12.0 m/s2 .	(71)

At this event the string is at a crest of the spatial profile and also at an instantaneous turning point of the local motion. Thus both slope and transverse velocity are zero, while the curvature and acceleration are negative.

Solution 7: Verify the wave-equation relation for a sinusoid

The wave has

(72)

1. (73)

2. Since

   (74)

   and

   (75)

   we obtain

   (76)

3. But

   (77)

   Therefore

   (78)

4. This calculation verifies that the proposed sinusoidal traveling wave satisfies the differential relation. It does not explain why a real stretched string must obey that equation or why its speed is set by the Tension and linear mass density. That physical derivation requires force balance and Newton’s second law.

Solution 8: Chain rule for an arbitrary right-moving profile

Let

(79)

The needed derivatives of ξ are

(80)

Then

ux	= F′(ξ)	(81)
	= F′(ξ),	(82)

and

ut	= F′(ξ)	(83)
	= −cF′(ξ).	(84)

Hence

(85)

Differentiate again:

(86)

and

(87)

Therefore

(88)

The derivation does not require F to be sinusoidal. It only requires the translating form and sufficient differentiability.

Solution 9: Left-moving profile and the sign change

Let

(89)

Then

(90)

Therefore

(91)

and

(92)

Thus

(93)

for a left-moving profile.

The second derivatives are

(94)

and

(95)

so once again

(96)

The first derivative contains one factor of the sign of the propagation term. Differentiating twice produces the square of that sign, so the second-order relation is the same for both directions.

Solution 10: Interpret the right-moving slope-rate relation

For a right-moving profile,

(97)

with c > 0.

1. If u_x > 0, then

   (98)

2. If u_x < 0, then

   (99)

3. At a fixed position, a right-moving profile brings material from slightly to the left into the observation point as time advances. If the spatial profile rises with increasing x, the values just to the left are smaller, so the local field decreases with time. This produces the opposite sign between u_t and u_x.
4. For a left-moving profile G(x + ct),

   (100)

Solution 11: Partial derivative versus total derivative

For a moving observer x = x(t), the measured quantity is u(x(t),t). The multivariable chain rule gives

(101)

For a right-moving profile,

(102)

Therefore

	= −cux + ux	(103)
	= ux.	(104)

Thus

(105)

when

(106)

An observer moving to the right at exactly the wave speed follows a fixed feature of the translating profile. Along that path, the value of F(x − ct) remains constant.

Solution 12: Mixed partial derivatives

Given

(107)

first differentiate with respect to x:

(108)

Differentiate with respect to t:

(109)

Now

uxt	= (2xt3 + 4t)	(110)
	= 6xt2 + 4 .	(111)

And

utx	= (3x2t2 + 4x)	(112)
	= 6xt2 + 4 .	(113)

Therefore

(114)

This agrees with equality of mixed partial derivatives for sufficiently smooth functions.

Solution 13: Diagnose conceptual statements

1. Incorrect. u_t is the local time rate of change at fixed position. For string displacement it is local transverse material velocity. The propagation speed of the wave pattern is a separate quantity such as c = ω∕k.
2. Correct. In u_x, position changes while time is held fixed.
3. Incorrect. u_xx = 0 means the local curvature measure is zero. The displacement itself can be nonzero.
4. Incorrect. For F(x − ct),

   (115)

   so for c > 0 the two derivatives have opposite signs whenever they are nonzero.

5. Incorrect. Showing that one waveform satisfies u_tt = c²u _xx only verifies compatibility with that PDE. A physical derivation is required to show why a particular medium obeys that equation and what determines c.

Solution 14: Synthesis problem

The wave is

(116)

1. By comparison with A cos(kx − ωt + ϕ),

   (117)

2. λ	= = = m ≈ 1.047 m ,	(118)
   T	= = = s ≈ 0.262 s ,	(119)
   f	= = = 3.82 Hz ,	(120)
   c	= = = 4.0 m/s .	(121)

3. At the event x = 0.10 m and t = 0.025 s, the phase is

   𝜃	= 6(0.10) − 24(0.025) +	(122)
   	= 0.6 − 0.6 +	(123)
   	= .	(124)

   Therefore

   (125)

   The spatial derivative is

   ux	= −Ak sin 𝜃	(126)
   	= −(0.015)(6)	(127)
   	≈−0.0779 .	(128)

   The time derivative is

   ut	= Aω sin 𝜃	(129)
   	= (0.015)(24)	(130)
   	≈ 0.312 m/s .	(131)

4. Since c = 4.0 m/s,

   (132)

   This equals u_t to rounding:

   (133)

5. First find the displacement at the event:

   u	= 0.015 cos	(134)
   	= 0.015	(135)
   	= 0.0075 m.	(136)

   Then

   uxx	= −k2u	(137)
   	= −36(0.0075)	(138)
   	= −0.270 m−1 ,	(139)

   and

   utt	= −ω2u	(140)
   	= −576(0.0075)	(141)
   	= −4.32 m/s2 .	(142)

6. c2u xx	= (4.0)2(−0.270)	(143)
   	= 16(−0.270)	(144)
   	= −4.32 m/s2.	(145)

   Therefore

   (146)

7. The quantity u_t is the transverse velocity of the material point at a fixed spatial label, while c is the speed at which the wave pattern propagates through the medium.

Common mistakes

• Mistake: differentiating both x and t when taking a partial derivative. Change one independent variable at a time.
• Mistake: confusing u_t with wave propagation speed. They describe different motions.
• Mistake: dropping the chain-rule factors k or ω when differentiating a sinusoidal phase.
• Mistake: forgetting the second minus sign when differentiating twice, leading to an incorrect sign in u_xx or u_tt.
• Mistake: using u_t = −cu_x for a left-moving profile. The left-moving relation is u_t = +cu_x.
• Mistake: treating a PDE verification as a physical derivation of the governing wave equation.

What WM13E1 reinforces

The exercises reinforce the interpretation of partial derivatives as directional rates of change through the (x,t) domain:

(147)

(148)

(149)

and, for transverse displacement,

(150)

For translating profiles,

(151)

and

(152)

while both satisfy

(153)

This completes the calculus preparation needed for the later physical derivation of the one-dimensional string wave equation.

References

References