GRE Physics Companion: Wave Equations in Two and Three Dimensions and Spherical Waves

GRE Physics Companion: Wave Equations in Two and Three Dimensions and Spherical Waves

The multidimensional scalar wave equation is

(1)

A plane wave satisfies

(2)

while an outgoing spherically symmetric wave has amplitude proportional to

(3)

1 High-value GRE facts

1. In Cartesian three dimensions, ∇² = ∂ _xx + ∂_yy + ∂_zz.
2. A plane wave has phase k ⋅ r − ωt.
3. Plane wavefronts are perpendicular to k.
4. k = 2π∕λ.
5. For a nondispersive wave equation, ω = ck.
6. In spherical symmetry, ∇²ψ = ψ _rr + (2∕r)ψ_r.
7. The substitution u = rψ gives the one-dimensional wave equation.
8. Spherical-wave amplitude falls as 1∕r.
9. Spherical-wave intensity falls as 1∕r².
10. Ideal cylindrical far-field intensity falls as 1∕ρ.

Part I: Original GRE-style problems

Problem 1: three-dimensional wave equation

Which equation is the scalar wave equation in Cartesian three dimensions?

1. ψ_x + ψ_y + ψ_z = cψ_t
2. ψ_xx + ψ_yy + ψ_zz = ψ_tt∕c²
3. ψ_xx + ψ_yy + ψ_zz = c²ψ _t
4. ψ_xx = ψ_tt∕c
5. ∇ψ = cψ

Problem 2: wave-vector magnitude

For

(4)

the magnitude of the wave vector is

1. 1 m⁻¹
2. 3 m⁻¹
3. 4 m⁻¹
4. 5 m⁻¹
5. 7 m⁻¹

Problem 3: wavelength

For the wave vector in Problem 2, the wavelength is closest to

1. 0.20 m
2. 0.80 m
3. 1.26 m
4. 5.00 m
5. 31.4 m

Problem 4: wavefront orientation

For a plane wave, surfaces of constant phase are

1. parallel to k
2. perpendicular to k
3. always spherical
4. always cylindrical
5. unrelated to k

Problem 5: spherical amplitude

An outgoing spherical-wave amplitude is A at radius r. At radius 2r, the ideal amplitude is

1. 4A
2. 2A
3. A
4. A∕2
5. A∕4

Problem 6: spherical intensity

If spherical-wave intensity is I at radius r, then at radius 2r it is

1. 4I
2. 2I
3. I
4. I∕2
5. I∕4

Problem 7: radial substitution

The substitution that converts the three-dimensional spherically symmetric wave equation into a one-dimensional wave equation is

1. u = ψ∕r
2. u = rψ
3. u = r²ψ
4. u = ψ
5. u = ψ_rr

Problem 8: outgoing spherical solution

Which form represents an outgoing spherically symmetric solution?

1. F(r + ct)
2. rF(r − ct)
3. F(r − ct)∕r
4. F(t)∕r²
5. F(r)∕t

Problem 9: cylindrical intensity

For an ideal cylindrical wave in the far field, intensity scales approximately as

1. constant
2. 1∕ρ
3. 1∕ρ²
4. 1∕
5. ρ

Problem 10: travel time

A spherical acoustic pulse travels 1000 m in air at 343 m∕s. The travel time is closest to

1. 0.343 s
2. 1.00 s
3. 2.92 s
4. 3.43 s
5. 343 s

Problem 11: harmonic spatial equation

Substituting

(5)

into the wave equation gives

1. ∇²Φ − k²Φ = 0
2. ∇²Φ + k²Φ = 0
3. ∇Φ + kΦ = 0
4. Φ_tt = 0
5. ∇²Φ = 0 for all ω

Problem 12: geometric spreading

Which statement is correct in a lossless medium?

1. All wave amplitudes must be constant with distance.
2. Spherical intensity can decrease even without absorption.
3. A spherical amplitude must fall as 1∕r².
4. Cylindrical intensity must fall as 1∕ρ².
5. Geometric spreading changes wave speed.

Part II: Complete worked solutions

Solution 1

In Cartesian three dimensions,

(6)

Thus

(7)

Answer: (B).

Solution 2

(8)

Answer: (D).

Solution 3

λ	=	(9)
	=	(10)
	≈ 1.26 m.	(11)

Answer: (C).

Solution 4

Constant phase means

(12)

at fixed time. These are planes perpendicular to

(13)

Answer: (B).

Solution 5

Spherical-wave amplitude scales as

(14)

Therefore doubling radius halves the amplitude. Answer: (D).

Solution 6

Spherical intensity scales as

(15)

Therefore doubling radius reduces intensity by a factor of four. Answer: (E).

Solution 7

The exact radial reduction uses

(16)

Answer: (B).

Solution 8

The outgoing solution depends on retarded radial coordinate

(17)

and includes the spherical spreading factor:

(18)

Answer: (C).

Solution 9

Power per unit source length spreads over a circumference proportional to

(19)

Thus

(20)

Answer: (B).

Solution 10

t	=	(21)
	≈ 2.915 s.	(22)

Answer: (C).

Solution 11

With

(23)

the spatial equation is

(24)

Answer: (B).

Solution 12

Lossless propagation can still show geometric spreading. For a spherical wave,

(25)

because the same power crosses increasing spherical area. Answer: (B).

2 GRE checklist

1. Identify dimension before writing the Laplacian.
2. For plane waves, use vector magnitude k, not a single component.
3. Wavefronts are Normal to k.
4. Use u = rψ for exact spherical radial reduction.
5. Separate amplitude spreading from intensity spreading.
6. Use 1∕r amplitude and 1∕r² intensity for spherical waves.
7. Use cylindrical 1∕ amplitude only as the far-field scaling.
8. Distinguish the time-dependent wave equation from the Helmholtz equation.

References

References