4.6 More about stationary waves on strings

Question 1

What type frequency brings about the 1st harmonic pattern of vibration? Describe its shape/pattern.

Answer 1

The first harmonic pattern of vibration is seen at the lowest possible frequency.
This has an antinode at the middle as well as a node at either end.

Question 2

What is the wavelength of the wave that forms the 1st harmonic?

Answer 2

λ,1 = 2L

Question 3

What is the equation used to find the frequency of the 1st harmonic?

Answer 3

f = c/λ,1 = c/2L

where c is the speed of the progressive waves on the wire

Question 4

What is the pattern and wavelength of the 2nd harmonic?

Answer 4

There is a node at the middle so the string is in 2 loops. 2 antinodes and 3 nodes.
The wavelength is λ,2 = L because each loop has length of half a wavelength.

Question 5

What is the frequency of the 2nd harmonic?

Answer 5

f,2 = c/λ2 = c/L = 2f,1

Question 6

What is the pattern and wavelength of the 3rd harmonic?

Answer 6

Nodes are at a distance of 1/3L from either end and an antinode at the middle. The wavelength is λ3 = 2/3L because each loop has a length of half a wavelength.

Question 7

What is the frequency of the 3rd harmonic?

Answer 7

f3 = c/λ3 = 3c/2L = 3f,1

Question 8

What is the pattern between loops and nodes?

Answer 8

Number of loops = number of nodes - 1

Question 9

Explain how a stationary wave is formed

Answer 9

Vibration generator creates a progressive wave in which reaches a boundary and gets reflected in the opposite direction. 2 waves are now interfering with each other in opposite directions with the same frequency and wavelength. At the nodes, there is destructive interference (0 amplitude) and at the anti nodes there is constructive interference (added amplitude of separate amplitudes from separate waves). / superposition of these 2 progressive waves generates nodes and antinodes.

Question 10

Describe an experiment used to produce stationary waves

Answer 10

1. A string or wire is tied at one end to a mechanical vibrator connected to a frequency generator.
2. The other end of the string passes over a pulley and supports a weight, which keeps the tension in the string constant

Question 11

How does tension of a string relate to pitch?

Answer 11

Raising the tension or shortening the length increases the pitch.

Lowering the tension or increasing the length lowers the pitch

Question 12

What is the equation used to find the first harmonic frequency?

Answer 12

f = 1/2l√T/µ
T = tension
µ = mass per unit length