Standing Waves on a Guitar String

[sphouxay]

Homework Statement

Learning Goal: To understand standing waves, including calculation of and , and to learn the physical meaning behind some musical terms.

The columns in the figure (Intro 1 figure) show the instantaneous shape of a vibrating guitar string drawn every 1 . The guitar string is 60 long.
The left column shows the guitar string shape as a sinusoidal traveling wave passes through it. Notice that the shape is sinusoidal at all times and specific features, such as the crest indicated with the arrow, travel along the string to the right at a constant speed.

The right column shows snapshots of the sinusoidal standing wave formed when this sinusoidal traveling wave passes through an identically shaped wave moving in the opposite direction on the same guitar string. The string is momentarily flat when the underlying traveling waves are exactly out of phase. The shape is sinusoidal with twice the original amplitude when the underlying waves are momentarily in phase. This pattern is called a standing wave because no wave features travel down the length of the string.



What is the wavelength of the standing wave shown on the guitar string?

Homework Equations

The Attempt at a Solution

I answered 120 cm, but it was not correct, someone please explain to me where I went wrong. Thanks

 

[Mark44]

You don't provide the information in the figure, so it's hard to know where the node is in the string. Assuming that there is a node in the exact middle of the string, the wavelength of each standing wave is half the length of the string, not twice the length of the string.

When you barely touch a guitar string at the 12th fret and pluck the string you get what is called a harmonic, a tone that is an octave higher than the natural tone of the unfretted string. You can also get another harmonic at the 7th fret, that is another octave higher. In this case you are setting up four standing waves, where the wavelength of each if 1/4 the length of the string.

By length of the string, I mean the distance from the nut to the bridge, and this length does not include the portion between the nut and the tuners or between the bridge and tailpiece or where the strings are fastened to the body.

 

[Mene]

The correct answer is actually 30 cm. This can be calculated using the equation λ = 2L/n, where λ is the wavelength, L is the length of the string, and n is the number of nodes in the standing wave. In this case, n = 2, since there are two nodes (points of zero amplitude) in the standing wave. Therefore, the wavelength is 2(60 cm)/2 = 30 cm.

It seems like you may have accidentally multiplied by 2 instead of dividing by 2 in your calculation. Remember that the wavelength is the distance between two consecutive nodes, so it should be half of the length of the string.