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BMAT Physics Notes: Waves

Advice & Insight From BMAT Specialists

​Introduction to Waves

The diagram shows a typical wave.

​Some basic definitions

CREST: The highest point on a wave.
 
TROUGH: The lowest point on a wave.
 
AMPLITUDE (A): The maximum displacement of the wave from the undisturbed position. (Measured in metres/cm).
 
WAVELENGTH (λ): The distance between two adjacent corresponding points. (Measured in metres/cm).
 
FREQUENCY (f): The number of oscillations/vibrations/cycles per second. (Measured in Hertz (Hz). [ 1 Hz = 1 s⁻¹]
 
PERIOD (T): The time for one vibration/oscillation/cycle of the wave. (Measured in seconds).
 
CYCLE: One complete ‘up and down’ part of a wave.

From the definitions it follows that:

 f  =  1/T     and     T = 1/f
[It is important to note that T MUST be in seconds for f to be in Hz]

Example 1
 
Calculate the frequency of a wave if the period is: 

(i) 5s __________    (ii) 0.02s _____________  (iii) 1 ms _____________  (iv)     20ms ____________       (v) 10μs  ____________

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​Example 2

Calculate the period of a wave if the frequency is: 

​(i) 4Hz ____________     (ii) 0.02Hz  ___________  (iii) 1kHz  ___________   (iv)  2 MHz ____________     (v) 0.5MHz __________

​The Wave Equation

It can be show that for ANY wave:            v  =  f λ

CARE! Do NOT mix the units for v and λ. They must  BOTH be in metres or BOTH in cm].
 
ALSO, note that the speed v of any wave is also given by:    speed (v) =  distance 
                                                                                                                        time


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​Example 3

Calculate the speed of wave if: 
(i) the frequency is 0.1kHz and wavelength 10 cm 
(ii) the frequency is 0.2 MHz and the wavelength is 100μm​

Example 4

Calculate the frequency of a wave if: 
(i) the speed is 100 ms⁻¹  and the wavelength is 0.5 m, 
(ii) the speed is 3 x10 ms⁻¹  and the wavelength is 1.5μm.  

Example 5

Calculate the wavelength of a wave if: 
(i) the speed is 0.2 ms-1 and the frequency is 0.5 Hz, 
(ii) the speed is 25 ms-1 and the frequency is 0.5 kHz. 

TRANSVERSE WAVE

If the particles of a wave vibrate at right angles to the wave direction the wave is said to be transverse.
Examples of transverse waves: ALL em waves (e.g. light, radio, X-rays…)
A TRANSVERSE wave can easily be shown by a slinky or rope as shown below.

LONGITUDINAL WAVE

If the particles of a wave vibrate parallel to the wave direction the wave is said to be longitudinal.
Example of longitudinal wave: Sound.
A LONGITUDINAL wave can best be shown by a slinky as shown below.

The centre of a ‘squashed’ region is called a compression (C).
The centre of a ‘stretched’ region is called a rarefaction (R).

Progressive wave

A progressive (or travelling) wave transfers energy through a medium without a net movement of matter.

Drawing Waves

There are two ways of drawing ‘waves’:
 
(a)  A ‘displacement-displacement’ graph.  This is a diagram of ALL of the particles along the wave at a FIXED time.
      (Like a ‘photograph’ of the wave).
 
(b) A ‘displacement-time’ graph.  This is a diagram of ONE particle along the wave at DIFFERENT times.
      [BOTH types of graph can be drawn for a transverse or a longitudinal wave].

​Example 6
 
The diagrams show a progressive wave. Displacement d is plotted on the vertical axis and either displacement (x) or time (t) is plotted on the horizontal axis. The period (T), wavelength (l) and amplitude (A) are also shown.
However, only one graph is correctly labelled.
 
Which graph is correctly labelled?

​[ANSWER: D]
The diagrams below refer to the same wave.
 
(a) Indicate on the relevant graph the amplitude, period and wavelength. 
 
(b) Use the graphs to calculate the speed of the wave.

Sound Waves

When an object e.g. a tuning fork vibrates it causes the surrounding air molecules to vibrate. These vibrations are passed on through the air. When the vibrations reach the ear drum it also is forced to vibrate at the same frequency. These vibrations are converted into an electrical signal to the brain and a ‘sound’ is heard.
 
If there was no air between the source (tuning fork) and the person then NO vibrations can be passed on. So it follows that sound needs a medium (solid, liquid or gas) to travel.
Sound travels fastest through a solid and slowest through a gas (air).
 
The human ear can detect sounds with frequencies from about 20 Hz to about 20 kHz.
The frequency of a sound wave is sometimes called the pitch.
The greater the amplitude of a sound wave the louder the sound.
Sound waves are longitudinal waves and can have wavelengths of about 20 cm.
When a sound wave is reflected from a surface an echo is formed.
 
Ultrasound are sound waves with frequencies greater then 20 kHz.
Typical values are in the MHz range.
Ultrasound can be used for sonar, medical imaging of body organs and measuring blood flow.

​The Doppler Effect

 Suppose a source emits a sound of constant frequency f If the source is stationary then two observers and will both hear the sound at a frequency f
 
This is shown in the first diagram.

​Suppose the source moves at a constant speed towards observer A. The crests of the wave from the source are squashed together in the direction of A and spread apart in the direction of B.
This is shown in the second diagram.

​The waves reaching observer A are arriving with a shorter wavelength and hence a higher frequency (remember the speed of the waves is constant).
The waves reaching observer B are arriving with a longer wavelength and hence a lower frequency.
Both observers hear sounds of different frequencies.
Person A will hear a higher frequency than f. Person B will hear a lower frequency than f.
The faster the source moves the greater the change in frequency detected. In this way the speed of a car or flow of blood can be measured.
This change in frequency (and hence wavelength) is called the Doppler effect.

​The Electromagnetic Spectrum

 (Small wavelength—LARGE FREQUENCY)
(LARGE WAVELENGTH – small frequency)

General Properties

  • They all travel at the speed of light (in a vacuum)
  • They are all transverse,
  • In general, the higher the frequency of the wave the more energy they carry and hence the more dangerous the wave.
  • X-rays and gamma rays  can damage human cells and cause cancer. This is because of their strong ionising ability.

LIGHT

REFLECTION

The diagram shows a light ray incident on a plane mirror and being reflected.
The dotted line is called a normal and is at an angle of 900 to the mirror.
The angle between the incident ray and the normal is called the angle of incidence (i).
The angle between the reflected ray and the normal is called the angle of reflection (r).

Experiment shows that :         i  =  r 

[Note: the speed (v), frequency (f) and wavelength (λ) do NOT change during reflection]

REFRACTION
 
When light travels from one medium (e.g. air) into another medium (e.g. water or glass) it changes direction.
This change in direction is called refraction.
This occurs because when light enters a different medium its speed changes.
If the speed decreases (e.g. light passing from air to glass/water) the light ray bends towards the normal.
If the speed increases (e.g. light passing from glass/water to air) the light bends away from the normal.
The angle between the refracted ray and the normal is called the angle of refraction (r).

​Note:

  • that the frequency (f) of the light does NOT change.
  • Since  v = f λ  if the speed (v) decreases then the wavelength (λdecreases.
  • If the speed (v) increases then then the wavelength (λincreases.
  • If a light ray travels along the normal then there is NO change in direction but v and λ still change.
  • Sunlight (i.e. ‘white’ light) is made of 7 colours: Red, Orange, Yellow, Green, Blue, Indigo and Violet.

          Although they all travel at the SAME speed in air, they travel at DIFFERENT speeds in glass/water.

  • This means each colour is refracted by different amounts.
  • RED travels the fastest in glass/water and is refracted the LEAST;
  • VIOLET travels the slowest in glass/water and is refracted the MOST.

Example Answers

​Example 1:   
(i) 5s f = 1/T = 1/5 = 0.2Hz       (ii) 0.02s  f = 50Hz (iii) 1 ms  f = 1000Hz     (iv)     20ms f = 50Hz      (v) 10μs  f = 10 Hz

​Example 2:   
(i) 4Hz T = 1/f =1/4 = 0.25s    (ii) 0.02Hz   T= 50s    (iii) 1kHz  T = 1 ms      (iv)  2 MHz  T = 0.5μs    (v) 0.5MHz T = 2μs

Example 3:  
(i) the frequency is 0.1kHz and wavelength 10 cm 
    v = fλ = 100×0.1 = 10 ms⁻¹

(ii) the frequency is 0.2 MHz and the wavelength is 100μm 
     v = 2 x 10 x100 x 10⁻⁶ = 20 ms⁻¹

Example 4:  
(i)  f = v/λ = 100/0.5 = 200Hz 

(ii) f = (3×10)/(1.5×10⁻⁶) = 2×10¹⁴Hz

Example 5:  
(i)  λ = v/f = 0.2/0.5 =0.4m 

(ii)  λ = 25/(500) = 0.05m 

Example 6:  
​(a) Indicate on the relevant graph the amplitude, period and wavelength.  
     Either graph for Amplitude, 1st graph for λ, 2nd graph for Period.

(b) Use the graphs to calculate the speed of the wave.  
      From 1st graph λ ≈ 3.4m  From 2nd graph  T = 0.2s  f = 5HZ   v = f λ = 5×3.4 = 17 ms⁻¹

BMAT Physics Notes: Waves

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