Standing Waves on Strings

Alex Law36551142

PHYS 101 LF2

Standing Waves on Strings

• Interference of two waves that move in the opposite direction– Consists of two harmonic waves with

equal amplitude, wavelength and frequency

• Standing waves on a string is created when a fixed string on both ends is plucked.– Results in travelling waves moving in

opposite directions

Standing Waves on Strings

• The wavelength of a standing wave is dependent on the length of the string, L and m which is a positive, nonzero integer:

• A smaller m gives the largest wavelength and a larger m gives a smaller wavelength. – This provides standing waves that are called

normal modes of vibration of the string

Standing Waves on Strings

• Frequencies corresponding to the normal modes of vibration are:

• A lower frequency will result in a larger wavelength and is known as the fundamental frequency or the first harmonic

Application

• Notes made from a piano are caused by varying lengths, and mass densities of the strings. On each opposite end of the keyboard, there will be a different mass density and length to produce either a high or a low frequency.

Application

Given the following information, determine the wave speed, wavelength, frequency (for the first harmonic) and conclude if the string resonates for a lower note or a higher note.• Length of string 1: 4.5m• Mass density of string 1: 2.49*10-2kg/m• Length of string 2: 1.5m• Mass density of string 2: 4.12*10-4 kg/m• The strings are held on a tension of 70.0N

Note: Values are arbitrary and are not accurate to what is actually found in a piano.

Solution

Wave speed for string 1:

Wavelength for string 1:

Frequency for string 1:

Solution (cont.)

Wave speed for string 2:

Wavelength for string 2:

Frequency for string 2:

Solution (cont.)

• Comparing the two frequencies:– f1 = 5.88Hz

– f2 = 13.74Hz

• A lower frequency results in a lower pitched sound and a higher frequency results in a higher pitched sound.

• It can be concluded that string 1 pertains to a lower note and string 2 pertains to a higher note. – This can be verified by looking at the wave speed

and wave lengths of each string.