Standing Waves On Strings: Harmonic and Frequency Problem

In summary, standing waves on strings are formed when two identical waves traveling in opposite directions interfere with each other. This creates stationary points and oscillating points on the string. These waves are created when a string is fixed at both ends and vibrated at a specific frequency. The frequency is determined by the tension and mass of the string and only certain frequencies will result in standing wave patterns. The harmonic number of a standing wave is directly proportional to its frequency, and musicians can manipulate these factors to produce different pitches in musical instruments. However, real-life factors such as damping, non-uniformity of string material, and inharmonicity can affect the formation of standing waves and alter the frequency and pitch of the sound produced.
  • #1
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Homework Statement


String A is stretched between two clamps separated by distance L. String B, with the same linear density and under the same tension as string A. String B is stretched between two clamps separated by distance 4L. Consider the first eight harmonics of string B. For which of these eight harmonics of B (if any) does the frequency match the frequency of (a) A’s first harmonic, (b) A’s second harmonic, and (c) A’s third harmonic?

Not sure if I correctly solved the problem (hopefully I did =D). Just need someone to check over my work =D. Thanks!

Homework Equations



ν = √(T/μ)

ν = ƒλ

L = nλ/2

ν: Velocity
T: Tension
μ: Linear Density
ƒ: Frequency
λ: Wavelength
L: Length
n: nth Harmonic
[/B]

The Attempt at a Solution


[/B]
Since the tension and linear density on both strings are equal, the velocity is also equal.

Next I solved for the frequency of system B:

ν = ƒλ
ƒ = ν/λ → 1

L = nλ/2, since L = 4L

λ = 8L/n → 2

Subbing 2 → 1

ƒ = νn/8L

--------------------------------------------------------------------------------
First Harmonic of String A:


L= λ/2 ⇒ λ = 2L
ƒ=ν/λ ⇒ ƒa1 = ν/2L

ƒa1 = ƒb

ν/2L = νn/8L
n = 4 (between 1-8)

Second Harmonic of String A:

L= λ
ƒ=ν/λ ⇒ ƒa2 = ν/L

ƒa2 = ƒb

ν/L = νn/8L
n = 8 (between 1-8)

Third Harmonic of String A:

L= 3λ/2 ⇒ λ = 2L/3
ƒ=ν/λ ⇒ ƒa3 = 3ν/2L

ƒa3 = ƒb

3ν/2L = νn/8L
n = 12 (not between 1-8)
 
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  • #2
Hello. Welcome to PF!

Your work looks correct.

A similar approach is to note that the harmonic frequencies of A are ##f_A = \frac{n_ {_A} \ v}{2L}## while those of B are ##f_B = \frac{n_{_B} \ v}{8L}##.

Show that ##f_A =f_B## implies ##n_{_B} = 4n_{_A}##. Then let ##n_{_A}= 1## for the first harmonic of A, etc.
 
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  • #3
Thanks for the help! I will try the question both ways =D.
 

FAQ: Standing Waves On Strings: Harmonic and Frequency Problem

1. What are standing waves on strings?

Standing waves on strings are patterns that form when two identical waves moving in opposite directions interfere with each other. This results in certain points along the string remaining stationary while other points oscillate with maximum amplitude.

2. How are standing waves on strings created?

Standing waves on strings are created when a string is fixed at both ends and vibrated at a specific frequency. This frequency is determined by the tension and mass of the string, and only certain frequencies will result in standing wave patterns.

3. What is the relationship between harmonic and frequency in standing waves on strings?

The harmonic of a standing wave on a string refers to the number of nodes (points of maximum amplitude) present on the string. The frequency of the wave is directly proportional to the harmonic number, meaning that as the harmonic increases, so does the frequency.

4. How do standing waves on strings affect the sound produced by a musical instrument?

Standing waves on strings are responsible for the production of different pitches in musical instruments. By altering the length, tension, or mass of a string, musicians can change the harmonic and frequency of the standing wave, resulting in a different pitch or note.

5. How do real-life factors affect standing waves on strings?

Real-life factors such as damping, non-uniformity of string material, and inharmonicity can affect the formation of standing waves on strings. These factors can cause the frequency of the standing wave to deviate from the ideal harmonic series, resulting in a slightly different pitch or sound.

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