Question

The G-string on a guitar is 64.8 cm long and is properly tuned to 196 Hz (fundamental) frequency.

1. Sketch the first harmonic on this guitar string at two different times.

2. What is the frequency of the G note one octave above this frequency? One octave below?

3. What is the frequency of the note a perfect fifth above this frequency?

4. What is the wavelength of the first harmonic of this guitar string?

5. What is the speed of waves on this guitar string?

6. If a tuning fork is played at 194 Hz, will there be a beat? At what frequency?

Answer 1

(1) Frquency of the G note one octave above 196 Hz = ( 196 x 2 ) Hz = 392 Hz

Frquency of the G note one octave below 196 Hz = ( 196 / 2 ) Hz = 98 Hz

(2) Let the frequency that is perfect fifth above 196 Hz be f

So, f : 196 = 3 : 2

or, f = ( 3 / 2 ) x 196 Hz = 294 Hz

(3) For first harmonic, length of the string = wavelength / 4

Hence, wavelength of first harmonic = Length of string x 4 = ( 64.8 x 4 ) cm = 259.2 cm

(4) Speed of wave = frequency of any harmonic x wavelength of that harmonic.

Let us assume first harmonic frequency and wavelength.

Hence, speed of wave in the string = ( 196 Hz x 259.2 cm ) = 5080.32 cm / s ~ 50.8 m / s

(5) If the difference between frequencies of any harmonic and the tuning fork is very less compared to their frequencies, then there will be a beat.

Now, all other frequencies, except fundamental, do not follow this one, since they have much higher frequencies.

So, there will be a beat when the guiter will be played in first harmonic.

Beat frequency will be : ( 196 - 194 ) Hz = 2 Hz