Good morning.
We've already learned about standing waves on a string, now we are going to apply that
concept to musical instruments.
♪flipping physics♪
Considering these are standing waves on a string, let's start with a stringed instrument
like a guitar, a violin, a piano and so on and so forth.
Stringed instruments use standing wave patterns to create frequencies which our brains interpret as pitch.
These instruments have strings which are fixed in place on either end.
Bobby, what does having ends which are fixed in place mean in terms of nodes?
- Can you bring back the animation which shows the nodes and anti-nodes on a standing wave pattern?
- Yes.
- Oh, thanks.
Well, a node is a location of tonal destructive interference and the net result is that the
string does not move at all at each node.
That must mean that the two ends of the strings are nodes on a stringed instrument, right?
- Correct.
Both of the ends of the strings on stringed instruments are nodes.
That means, the first standing wave pattern we can create on a string has two nodes, one
on each end and one anti-node in the middle.
When this is the case, Bo, how many wavelengths equal capital L, the length of the string?
- Well, that standing wave pattern represents half a wavelength, so half a wavelength equals
capital L, the length of the string.
- Right.
There for, the wavelength of this wave equals two times the length of the string.
Recall that the speed of a way of equals frequency times wavelength.
That means the frequency of the wave on the string will equal the speed of the wave divided
by the wavelength of the wave.
Or for this standing wave pattern, the frequency of the wave equals the speed of the wave divided
by the quantity two times the length of the string.
- Mr. P?
- Yes Bobby.
- What do you mean by the speed of the wave?
- Remember, the standing wave pattern is created by the constructive and destructive interference
of the periodic waves which are moving through and being reflected in the string.
- Yeah and we found the speed of a single wave pulse on a string, so that is the speed
he is talking about here, the speed of the wave pulse on the string.
- That's right, thanks.
-You're welcome.
- Billy, the next possible standing wave pattern has how many nodes and anti-nodes?
- Well, that would be three nodes and two anti-nodes.
In that case, the wavelength of the wave equals the length of the string and the frequency
of the wave in the string equals the speed of the wave divided by the length of the string, right?
- Correct Billy.
And let's add one more...
Bobby, what about the next possible standing wave pattern?
- The next possible wave pattern has four nodes and three anti-nodes.
That means there are 1 1/2 wavelengths on the length of the string and the wave length
equals 2/3 the length of a string and that means the frequency of the wave in this standing
wave pattern equals 3 times the speed of the wave divided by 2 times the length of the string.
- Very nice Bobby.
Now, I know it might not seem like there is a pattern, however, there is.
I think you will see it better when I algebraically manipulate the first frequency to equal 1
time speed divided by 2L.
The second frequency to 2 times speed divided by 2L and the third frequency to 3 times speed
divided by 2L.
- Each frequency is an integer times the quantity speed divided by 2 times the length of the string.
- Correct Bobby.
The frequency of the sound created by a stringed instrument equals the variable lowercase n
times the speed of the wave on the string divided by the quantity 2 times the length
of the string.
The variable n is called the "Harmonic Number" and when n equals 1, the frequency is called
the "Fundamental Frequency" - Got it.
The fundamental frequency is when the harmonic number equals 1 and the standing wave pattern
has the lowest frequency, the largest wave length and the least number of nodes and anti-nodes
in the standing wave pattern on the string.
- The fundamental frequency is also called the first harmonic because the harmonic number equals 1.
When n, the harmonic number equals 2, that frequency is called the second harmonic.
When the harmonic number equals 3, that frequency is called the...
- Third harmonic.
When the harmonic number equals four, that frequency is called the...
- Fourth harmonic.
- When the harmonic number equals five, that frequency is called the...
- Fifth harmonic.
- I think we get it.
- Sure, but did you notice that each of the harmonics is the harmonic number times the
fundamental frequency?
- Oh right, that means each of the harmonics is an integer multiple times the first harmonic
or the fundamental frequency, and that's pretty cool.
- Let's take a moment to look at and hear what these frequencies sound like.
When I pluck the open A string of a guitar and we hear a 110 Hz note.
because, 110 Hz is the fundamental frequency for this string.
When I hold my finger against the middle of the string but do not press the string against the fretboard,
I force a node at that point in the string and I can pluck the string to create the second harmonic.
What is the frequency of this standing wave pattern?
- It is the second harmonic, so it should be two times the fundamental frequency.
Right, so it is 2 times 110 Hz or 220 Hz.
- Correct.
And if I again hold my finger against the string but do not press the string against
the fret board, I force a node at this point.
This time I'm holding my finger one-third of the way from the end of the string.
Bo, what is going on now?
Well, if you are creating a node at one third L,
then this should be the third harmonic or three times 110 or 330 Hz.
What if you move your finger to the other end of the string
and force a node again one third from the end of the string?
Bobby, that forces a node at one third L, which will create the same third harmonic at 330 Hz.
What about all the other strings on the guitar? They create different frequencies than that A string,
however, they all have the same length, right?
Right, each of the strings creates a different frequency or pitch. I am not going to get into the specifics of why
in this video, however, I will say that the mass per unit length of a string
affects the speed of the wave on the string.
A larger mass per unit length means a smaller speed and therefore a smaller frequency.
That is why the lower pitch strings are larger in diameter to have a larger mass per unit length
and therefore a smaller speed and a smaller frequency.
Also, turning the tuning pegs of the guitar adjusts the tension in the strings
and an increased tension means an increased wave speed and an increased frequency.
Thank you very much for learning with me today, I enjoyed learning with you.