M4 Stationary Waves

Question 1

when do stationary waves occur

Answer 1

when two waves of the same frequency and amplitude, travelling in opposite directions in the same region as each other interfere

Question 2

give examples of stationary waves

Answer 2

stretched springs
air columns
stringed instruments like guitars

Question 3

what can you observe in the meldes experiment (wave on a spring)

Answer 3

• there are points on the stationary wave where the displacements of the particles are zero - these points are called nodes
• between one node and the next all particles oscillate in phase with each other but with different amplitudes - point of greatest amplitude is the antinode
• the oscillations of the particles in one loop are in antiphase with those in the next loop
• the frequency and wavelength of the stationary wave = the frequency and wavelength of the progressive wave

Question 4

draw the pattern observed of the stationary wave when the length is 1.5λ and when the length is 1λ

Answer 4

for the 1.5λ there are 4 nodes and 3 antinodes

for 1λ there are 3 nodes and 2 antinodes

Question 5

how can you investigate stationary waves in microwaves

Answer 5

microwaves are incident on a metal plate where they are reflected
place a point microwave detector along the wave, it will register signals every half wavelength. can detect nodes and antinodes

at the node - amplitude of vibration = 0 so no microwaves so 0 reading on detector

at antinodes - maximum amplitude so high intensity of microwaves so max reading on detector

Question 6

what are the similarities between progressive and stationary waves

Answer 6

same wavelength

same frequency

Question 7

what are the differences between progressive and stationary waves

Answer 7

• progressive waves transfer energy from one place to another whereas stationary waves store energy
• in progressive waves each particle has the same amplitude but they are out of phase with each other whereas in stationary waves particles have different amplitudes but within a ‘loop’ they are in phase with each other

Question 8

what is the frequency of the second harmonic of a string

Answer 8

L = λ

sub into v= fλ
v = fL
so f1 = v / L

Question 9

what is the frequency of the third harmonic of a string

Answer 9

L = (3λ) / 2 
wavelength = 2L / 3

sub into v = fλ
v = 2fL / 3

f2 = 3v / 2L

Question 10

compare the frequencies of modes of vibration on a stretched spring

Answer 10

f1 = 2f0
f2 = 3f0

the frequencies are in the ratio 1:2:3 - harmonic series

Question 11

in a closed pipe what is the closed end and what is the open end

Answer 11

closed end - node

open end - antinode

Question 12

draw the fundamental mode of vibration for a closed pipe

Answer 12

L = λ / 4
λ = 4L
v = f4L

f0 = v / 4L

Question 13

draw the second mode of vibration for a closed pipe

Answer 13

L = 3λ / 4
λ = 4L / 3
v = f X 4L / 3
f1 = 3v / 4L

Question 14

draw the third mode of vibration of a closed pipe

Answer 14

L = 5λ / 4
λ = 4L / 5
v = fλ
v = f X 4L / 5
f2 = 5v / 4L

Question 15

compare the frequencies of the modes of vibration in a closed pipe

Answer 15

f1 = 3f0
f2 = 5f0

ratio is 1:3:5

Question 16

in a closed pipe what happens when a stationary sound wave travels down it

Answer 16

it reflects at the bottom
resonance occurs - sound appears louder

because there is also an antinode at the open end as there is a node at the closed end

Question 17

how do you present stationary sound waves in a closed tube as longitudinal

Answer 17

the node has no arrows as there’s no vibration
the next particle at the antinode has an up and down arrow as there is max vibration

so every other particle has arrows with the first one at the bottom not having any

Question 18

what experiment could you use to determine the speed of sound

Answer 18

use a closed tube and a tuning fork

by hovering the fork above the closed tube you can see the length where there is the loudest sound
times the length by 4 to get the wavelength as you worked out the antinode
v = fλ

it’s around 343 ms^-1

Question 19

what’s the alternate method for determining the speed of sound
(finding 2 resonances)

Answer 19

the first resonance you hear is L1
the second resonance you hear is L2
L1 = λ / 4
L2 = 3λ / 4

L2 - L1 = λ / 2 so 2(L2 - L1) =λ
frequency is known from the tuning fork so speed of sound can be found:
v = fλ
v = f2(L2 - L1)

Question 20

what is the period of a wave

Answer 20

the time it takes for 1 complete wave to pass a point

Question 21

on an oscilloscope what is along the x axis

Answer 21

the timebase

Question 22

if the timebase was 1ms / div and the period was over 4 squares, what is the frequency of the wave

Answer 22

f = 1 / T
T = 4 X 10^-3

f = 1 / 4X10^-3 = 250 Hz

Question 23

if the timebase was 3ns / div and the period was over 8 squares, what is the frequency of the wave

Answer 23

T = 3X10^-9 X 8 = 24 X 10^-9

f = 1 / 24 X 10^-9
= 4.2 X 10^7 Hz

Question 24

what is the end correction

Answer 24

the vibrations within the tube will be transmitted to the air just outside the tube and the air will also vibrate
this means we consider that the tube is effectively longer than its measured length by an amount d

the equation for the closed tube then becomes f = v / 4(L + d)

Question 25

what type of wave are stationary waves? so how do the particles vibrate?

Answer 25

transverse

each particle vibrates at right angles to the spring

Question 26

what’s resonant frequencies

Answer 26

when an exact number of half wavelengths fits on the string

Question 27

what is the fundamental mode of vibration (first harmonic) of a string

Answer 27

L = λ/2
v = fλ
v / 0.5L = f
f0 = v/2L

Question 28

a banjo string vibrates with a first harmonic frequency of 290 Hz.
find the frequency of vibration of the string at the third harmonic

Answer 28

the third harmonic is three times the frequency of the first harmonic
f = 290 X 3 = 870Hz