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WAVE MOTION

 

Waves in Strings

 

fundamental

harmonics

frequency theory

mass, tension, length

 

 

 

The Fundamental Frequency

 

If we consider a length of string with one end tethered, a wave can be sent from the other end by waving the string up and down. The wave reflects at the tethered end and proceeds in the opposite direction.

 

Consider now a continuous wave being produced. The wave travelling to the left interferes with the reflected wave moving to the right.


In this way 'standing waves' are set up. The Fundamental Frequency is simply the lowest frequency for a standing wave to form.

 

 

the Fundamental Frequency

 

 

The Fundamental Frequency is just one of a series of particular frequencies called overtones or harmonics, where standing waves form.

 

 

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Harmonics and Overtones

 

 

the second harmonic

 

 

The Fundamental Frequency is called the 1st harmonic.

 

Successive frequencies where standing waves are produced are called the 2nd harmonic, the 3rd harmonic and so on.

 

 

the 3rd harmonic

 

 

Similarly, the higher frequencies above the Fundamental are termed overtones.

 

The next highest frequency above the Fundamental is called the 1st overtone. The next highest the 2nd overtone etc.

 

So,

the 2nd harmonic is the 1st overtone.


the 3rd harmonic is the 2nd overtone

etc.

 

From the diagrams it can be seen that there is a pattern connecting the wavelength(λ) and the length of the string (L).

 

harmonics equations

 

For the nth harmonic, the wavelength λn is given by:

 

 

general harmonic equation

 

 

 

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Frequency theory

 

First, let us use the familiar wave equation linking velocity, wavelength and frequency,

 

wave velocity in terms of wavelength and frequency

 

Hence the frequency of the nth harmonic ( fn ) is given by:

 

frequency of the nth harmonic                              (i       

 

where λn is the wavelength of the nth harmonic and v is the velocity of the wave in either direction.

 

From the previous section, the wavelength λn is given by:

 

general harmonic equation

 

hence,

 

wavelength of the nth harmonic

 

 

substituting for 1/λn into equation (i

 

frequency of the nth harmonic - lambda substituted                             (ii        

 

With n=1 , frequency of the 1st harmonic (the Fundamental) f1 is given by:

 

fundamental frequency in terms of wave velocity and string length

 

 

Substituting for v/2L into equation (ii , we obtain the frequency of the nth harmonic in terms of the Fundamental frequency.

 

harmonics as multiples of the fundamental frequency

 

Thus proving that subsequent harmonics are all multiples of the Fundamental Frequency.

 

 

 

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Effect of mass/unit length, length, tension on frequency

 

By experiment, it can be shown that,

 

wave velocity in terms of string tension and mass per unit length                             (iii       

 

where,


T is the tension in the string - Newtons (N)
μ is the mass/unit length of the string - (kgm-1)

From equation (ii above,

< > 

frequency of the nth harmonic - lambda substituted

 

making v the subject,

wave velocity

 

substituting for v from equation (iii above,

derivation of harmonic frequency in erms of L, T and mu

making fn the subject,

 

 

harmonic frequency in terms of string length, tension and mass per unit length

 

 

From the equation it can be seen that:

 

waves in strings - proportionalities

 

 

The proportionalities are often termed the Laws of Vibration for Stretched Strings.

 

In simple terms,

 

long strings make low frequencies and vice versa;

 

tight strings make high frequencies and vice versa;

 

thick, heavy strings make low frequencies and vice versa.

 

 

 

 

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