# Vibration on the guitar when playing two strings

I know that when I am playing one string on the guitar, it creates a standing wave which causes the entire body vibrates in its frequency and therefore create sound waves. But, what about two strings?

Do the two waves cause the same body to vibrate in some kind of a sum of the two frequencies? How come you get the sound of both from the same body?

TNX

• Standing waves are only in the strings, two different standing waves in two separate strings. The oscillations of the top engaging the air inside and outside the body are not standing waves, because the body is not a resonator. The top acts as a dipole transformer of the acoustic impedance between the strings and the air; the body is a phase inverter for the back wave of the dipole. Please see these answers: physics.stackexchange.com/questions/452833/… - and: physics.stackexchange.com/questions/365557/365569#365569 – safesphere Aug 7 '19 at 6:59

The first image shows a string oscillating at its fundamental frequency $$f$$.

The second image shows a string oscillating at $$2f$$.

The third image shows a string (or the wooden soundbox of a guitar or the air in and around a guitar) oscillating at both frequencies at the same time. Finally, here is a graph showing the height of a point a short way along the last string, as a function of time. Notice that it vibrates with an overall period of $$2\pi$$, but within each period there is also a faster oscillation with period $$\pi$$. The body of the guitar will vibrate at both frequencies at the same time.

This is because the body of the guitar is approximately a linear system so the vibration at each frequency is independent of each other.

This is also the case for air, which is why we can hear multiple things at the same time.

• What about the linear system that makes this one happen? I just can't imagine how a guitar body can vibrate in different frequencies. Do some parts are related to some frequencies? And sound waves sometimes interfere destructively. – Ofir Shukrun Aug 6 '19 at 12:45
• think of a single radio speaker, it can vibrate with all the sounds of multiple instruments and voices at the same time – Adrian Howard Aug 6 '19 at 13:05
• @Ofir Shukrun try waving your hand at your wrist and moving your arm side to side at the same; your hand is vibrating at two frequencies – user234190 Aug 6 '19 at 13:07
• Thanks for the answers, I will give this topic a second reading. – Ofir Shukrun Aug 6 '19 at 13:56

Do the two waves cause the same body to vibrate in some kind of a sum of the two frequencies?

Yes. The vibration state of the guitar with two strings is a superposition of the two string frequencies. This state doesn't have a well defined frequency.

How come you get the sound of both from the same body?

The guitar body isn't generating two sounds. Its generating just one-the superposition state. Its the ears which are able to discern the two tones being played simultaneously.

Further explanation:

In a simple picture$$^2$$, the vibrations from the two strings add up(in the guitar body) in the manner $$\alpha A_{1}(t)+\beta A_{2}(t)$$ where $$A_{i}(t)$$ is the vibration amplitude at time $$t$$ of the $$i^{th}$$ string. This is what we mean by linear superposition.(This is they way sound "adds" at low intensities). Here the constants(in time) $$\alpha$$ and $$\beta$$ are in general real$$^1$$. This is important. They are determined by the way the strings were stroked, the tensions in the strings, the dissipation environment, string-guitar coupling etc. But once the strings are stroked, they determine the contribution to the whole sound from each individual tone.

The guitar is now vibrating in a state which follows the above description. Is this state periodic with some frequency? Not really. For general coefficients and general ratio of the two string frequencies, this state doesn't have a well defined frequency--after no amount of time does the signal repeat. The notion of frequency isn't useful here. There is an instantaneous frequency- though its not the frequency in the usual sense.

To re-emphasize-the guitar isn't in two states of well defined frequencies vibrating simultaneously-it is in just one state-the superposition- of ill-defined frequency.

Why are we able to hear the two sounds? I am not entirely sure here but its the brain which is able to discern the two frequencies. There is an obvious aural resolution involved. Again the eardrum doesn't vibrate at two frequencies-it receives the superposition only.

1. In general they can be time dependent and complex.
2. A more complicated picture is that the vibrations (comprising of contributions from more than just the fundamental harmonics of both strings) excite the normal modes of the guitar cavity.
• So basically, the superposition is happening in the body of the guitar and not on the strings themselves? (like in the double-slit experiment when the waves interfere with each other) – Ofir Shukrun Aug 7 '19 at 13:59
• simply speaking yes. In theory superposition happens in all space where there are sound waves from the two strings-the air around the strings, the air around the guitar, the walls of the room, your eardrums, etc. – lineage Aug 7 '19 at 14:01
• Indeed. Thank you :) – Ofir Shukrun Aug 7 '19 at 14:04