# standing waves in strings

The length of the smallest string on a piano is about .15 m long and has a frequency of 4186 Hz. The lowest note on a piano has a frequency of 27.5 Hz. If you were making a piano with the same thickness wire (same µ) for each note, and keeping the tension the same for each note, how long would the string for the lowest note of the piano be and the calculations/formula.

standing waves in strings:

frequency = (1/(2 length of string))*sqrt(Tension / mass density)

Let's use some simpler numbers to illustrate the problem. Let's say the smallest string on a piano is `1m`, and has a frequency of `200hz` (it's a piano for giants). And let's say the lowest note on a piano is `100hz` (only one octave because our giants have a narrow hearing range).

Given all other factors are consistent (tension, thickness, etc...), we know the frequency is inversely proportionate to the size of the string such that a string that is twice as long yields half the frequency:

``````[ 200hz highest note ]   /   [ 100hz lowest note ] = 2

longest string = [ 1m shortest string ] * 2 = 2m
``````

We could illustrate our logic with an equation:

``````200hz / 100hz = quotient of 2
1m * your quotient of 2 = 2m
``````

In other words, the formula is:

``````longestLength = shortestLength * (highestFrequency / lowestFrequency)
``````

So,

``````longestStringLength = 0.15m * (4,186Hz / 27.5Hz)
``````

The misleading thing is the provided equation:

``````frequency = (1/(2 length of string))*sqrt(Tension / mass density)
``````

Since tension, mass density, and the length of string is not provided, this equation isn't much help to us. Are you sure this is the equation you're supposed to use?