GRE Physics Companion: Continuous Strings, Standing Waves, and Normal Modes

GRE Physics Companion: Continuous Strings, Standing Waves, and Normal Modes

For a uniform stretched string,

(1)

For a string fixed at both ends,

(2)

and

(3)

1 High-value GRE facts

1. c = .
2. An ideal string satisfies y_tt = c²y _xx.
3. Fixed-fixed modes have k_n = nπ∕L.
4. Fixed-fixed frequencies are f_n = nc∕(2L).
5. The allowed wavelength is λ_n = 2L∕n.
6. Mode n has n − 1 internal nodes.
7. Distinct sine modes are orthogonal on 0 ≤ x ≤ L.
8. Arbitrary initial shapes are Fourier sine series.
9. A midpoint triangular pluck excites only odd harmonics.
10. A fixed-free string has only odd harmonic multiples of its fundamental.

Part I: Original GRE-style problems

Problem 1: string wave speed

For tension

(4)

and linear density

(5)

the wave speed is

1. T∕μ
2. 
3. 
4. Tμ
5. 1∕

Problem 2: fixed-fixed fundamental

The fundamental frequency of a fixed-fixed string is

1. c∕L
2. c∕(2L)
3. 2c∕L
4. πc∕L
5. c∕(4L)

Problem 3: third-mode wavelength

For a fixed-fixed string, the wavelength of the third normal mode is

1. L∕3
2. 2L∕3
3. L
4. 3L∕2
5. 6L

Problem 4: internal nodes

The fourth fixed-fixed normal mode has how many internal nodes?

1. 1
2. 2
3. 3
4. 4
5. 5

Problem 5: dispersion

For an ideal uniform string,

1. ω ∝ k²
2. ω = ck
3. ω is independent of k
4. v_g = 0
5. only discrete traveling waves exist on an infinite string

Problem 6: orthogonality

For

(6)

(7)

equals

1. 0
2. L
3. L∕2
4. 1
5. nm

Problem 7: midpoint pluck

A perfectly symmetric triangular midpoint pluck excites

1. only even harmonics
2. only odd harmonics
3. only the fundamental
4. no harmonics
5. equal amplitude in every harmonic

Problem 8: fixed-free fundamental

A string fixed at one end and free at the other has fundamental frequency

1. c∕(2L)
2. c∕L
3. c∕(4L)
4. 2c∕L
5. zero

Problem 9: free-end boundary condition

The ideal transverse free-end condition is

1. X(L) = 0
2. X′(L) = 0
3. X′′(L) = 0
4. Ẋ(L) = 0
5. X(L) = 1

Problem 10: numerical fundamental

For

(8)

the fundamental frequency is closest to

1. 59.6 Hz
2. 119 Hz
3. 155 Hz
4. 238 Hz
5. 357 Hz

Problem 11: modal coefficient

For initial displacement

(9)

and zero initial velocity, the fixed-fixed coefficient is

1. ∫ ₀^Lf(x) dx
2. ∫ ₀^Lf(x) sin(nπx∕L) dx
3. ∫ ₀^Lf(x) dx
4. ∫ ₀^Lf(x) cos(nπx∕L) dx
5. always zero

Problem 12: modal dynamics

Each fixed-string modal coordinate obeys

1. Q_n = 0
2. Q_n + ω_n²Q _n = 0
3. Q_n + ω_nQ_n = 0
4. Q_n − ω_n²Q _n = 0
5. a nonlinear equation necessarily

Part II: Complete worked solutions

Solution 1

The string equation gives

(10)

Therefore

(11)

Answer: (B).

Solution 2

For

(12)

(13)

Answer: (B).

Solution 3

(14)

Thus

(15)

Answer: (B).

Solution 4

Mode

(16)

has

(17)

internal nodes. Therefore mode 4 has 3. Answer: (C).

Solution 5

For the ideal string,

(18)

Both phase and group velocity equal

(19)

Answer: (B).

Solution 6

Distinct sine modes are orthogonal:

(20)

Answer: (A).

Solution 7

For a midpoint triangular pluck,

(21)

This vanishes for every even

(22)

Answer: (B).

Solution 8

The fixed-free allowed wave numbers are

(23)

For

(24)

(25)

Answer: (C).

Solution 9

Zero transverse force at an ideal free end requires zero slope:

(26)

Answer: (B).

Solution 10

c	=	(27)
	≈ 154.9 m∕s.	(28)

Then

f1	=	(29)
	≈ 119.2 Hz.	(30)

Answer: (B).

Solution 11

Orthogonality gives

(31)

Answer: (B).

Solution 12

Each normal coordinate is an independent harmonic oscillator:

(32)

Answer: (B).

2 GRE checklist

1. Identify whether the problem gives linear density μ or total string mass.
2. Compute c = before applying endpoint conditions.
3. For fixed-fixed strings, use L = nλ_n∕2.
4. Count internal nodes as n − 1.
5. Distinguish traveling-wave speed from material-point transverse speed.
6. Use sine-series projection for fixed-fixed endpoint data.
7. For a fixed-free string, use zero displacement at the fixed end and zero slope at the free end.
8. Remember that each normal coordinate obeys its own SHO equation.

References

References