Waves Cheatsheet

Waves

We can categorize waves according to their propagation direction under two title; longitudinal waves and transverse waves.

Transverse Wave: In these types of waves, directions of wave and motion of particles are perpendicular to each other. Picture given below shows this wave type.

Longitudinal Wave: In this type of waves, direction of the particles and wave are same. Look at the given picture below.

Pulse:  one wave motion created at the spring.

Where; x is the pulse length and y is the amplitude (height of the pulse).

Wavelength:It is the distance between two points of two waves having same characteristics.

Wavelength is shown with the Greek letter "" and unit of it is "m". 

 Period: Time required for production of one wave is called period. It is shown with letter "T" and its unit is "s". 

Frequency: It is the number of waves produced in a given unit of time. It is shown with letter "f" and its unit is "1/s".

Periodic Wave: If the wave source produces equal number of waves in equal times, then this wave called periodic wave.

 f=1/T

Velocity of the Wave: Velocity of the wave is constant in a given medium. However, if the medium is changed then velocity of the wave is also changed.  We show velocity with "v" and its unit is m/s.

Velocity of the Spring Pulse

Distance taken at an instant time by the pulse is called velocity of the pulse. Velocity of the spring pulse depends on force exerted on the spring and spring constant µ. Spring constant depends on the type of the spring; it is found by the following formula;

 µ=mass/length

Where; v velocity of the spring pulse (m/s), µ is the spring constant (kg/m), and F is the force exerted on the spring (N).

Interference of Spring Waves

Reflection of Spring Waves

a) Reflection from Fixed End:

When a pulse of spring wave hits an obstacle having fixed end, it reflects. Reflected wave has opposite direction, same amplitude and velocity with the incident wave. Picture given below shows this process.

b) Reflection from Open End :

When a pulse of a spring comes to an open end obstacle it reflects, like given picture below. Amplitude, velocity and length of the pulse do not change but its right hand side becomes left hand side.

Reflection from High Density to Low Density and Low Density to High Density:

Relation between the velocities of incident transmitted and reflected wave;

 vincident=vreflected>vtransmitted

Amplitudes of the incident wave and reflected wave are equal but amplitude of transmitted wave is larger than them. Relation between the velocities of incident, reflected and transmitted waves;

vincident=vreflected<vtransmitted

Refraction of Waves

Waves change direction when passing from one medium to another. This change in the direction of wave is called refraction of wave. During refraction velocity and wavelength of waves change however, frequency of waves stay constant. Velocity and wavelength of wave coming from deep part of water tank to shallow part decrease. We can write following equations for incident and refracted waves;

Refraction of Waves

Waves change direction when passing from one medium to another. This change in the direction of wave is called refraction of the wave. During refraction velocity and wavelength of waves change, however, the frequency of waves stay constant. Velocity and wavelength of the wave coming from deep part of the water tank to shallow part decrease. Picture given below shows this change in the velocity and wavelength of waves.

If a direction of waves coming from deep part of the water tank is not perpendicular to the normal of the surface, then directions of refracted waves change. Look at the given picture below, it shows incident wave, refracted wave and angles between them.

We can write following equations for incident and refracted waves;

Example: Velocity of the wave in the deep part of the tank is 16m/s. Picture given below shows the refraction of this wave when it pass to the shallow part of the tank. Find the wavelength of this wave in a shallow part of the tank.

Example: If the side view of the water tank is given in the picture below; draw the top view of the periodic linear waves produced by the source.

There are  three different depths in this tank. Thus, velocity and wavelength of the deep part are bigger than the velocity and wavelength of the shallow part.

  Example: If the top view of the water tank is given below, find the shape of the linear wave after refracting from the deep part of the tank. 

First two ends of the linear wave enter to the deep water, so the velocity of these points increase and linear wave becomes the circular wave.

Water waves

Glass water tanks are used for examining water waves and its properties. With the help of refraction of light properties of water waves will be explained. You can produce two types of water waves, circular and linear. Picture given below shows how we use light and determine the avelength of the water waves.

Crests of the waves behave like converging lens and we see at the bottom of the tank lightened area. On the contrary, troughs make us see darken area at the bottom of the water tank.

We can create linear waves using a rod, and circular waves using a point source.

 Reflection of Linear Water Waves

Linear waves reflect from a linear surface with an angle equal to incident angle. Pictures given below show the incident wave and reflected wave with their angles.

Linear waves reflect from a circular surface like in the given picture below. After reflection their shape becomes the reflection surface's shape.

In the first picture, waves converge at one point and they turn into circular waves like in the concave mirrors. However, in the second picture, waves reflect from the circular surface as if they are coming from a point behind the surface like in the convex mirrors. Linear waves in the second picture also turn into circular wave.

Reflection of circular wave can be explained like reflection of light from curved mirrors. Pictures given below shows some of the reflection of circular waves.

Picture given above shows the reflection of circular wave from concave surface. In this picture, waves come from the center of the curvature.

Picture given above shows the reflection of circular wave from concave surface. Waves come from the focal point of the surface and reflected circular waves become linear waves.

Reflection of Spring waves

Reflection of Spring Waves

 Hitting an obstacle and turning back of the wave is called a reflection of the wave. We examine reflection of spring waves under two title; reflection from a fixed end and reflection from an opened end.

a) Reflection from Fixed End:

When a pulse of spring wave hits an obstacle having fixed end, it reflects. Reflected wave has opposite direction, same amplitude, and velocity with the incident wave. Picture given below shows this process.

b) Reflection from Open End:

When a pulse of a spring comes to an open end obstacle it reflects, like given picture below. Amplitude, velocity, and length of the pulse do not change but its right-hand side becomes left-hand side.

Reflection from High Density to Low Density and Low Density to High Density:

We add two springs having different thicknesses and send a pulse from the spring having low density to high density. Some part of the pulse is transferred to the high density spring and continues its motion and rest of the pulse reflects. Joining point of two springs behaves like fixed end obstacle. Picture given below shows the behavior of incident, transferred and reflected wave.

Relation between the velocities of incident transmitted and reflected wave;

 vincident=vreflected>vtransmitted

 When a pulse send from the high density spring to low density spring, some part of the pulse reflects again and some part of it is transmitted. Picture given below shows the behavior of reflected and transmitted pulse.

Amplitudes of the incident wave and reflected wave are equal but amplitude of transmitted wave is larger than them. Relation between the velocities of incident, reflected and transmitted waves;

vincident=vreflected<vtransmitted

 Example: Draw the transmitted and reflected wave of the given pulse below.

Example: Find the directions of the pulses A and B if the directions of the particles x and y given below.

Given picture below shows the pulse A, and its shape after t second. From the positions of points x and y, we can say that this pulse travels along (-) direction and pulse B travels along (+) direction.

 Example: There are two waves having equal length and amplitude like in the given picture below. Find the shape of the waves when they overlap.

When two pulse overlap their shape becomes;

Interference of Spring Waves

Velocity of Periodic Waves

We have learned that, if the medium is constant than velocity of the wave is also constant and we gave following equations for velocity of waves;

Example:  Find the relation of wavelengths of given waves.

We define wavelength as the distance between the sequential crests or troughs.Picture given below shows the wavelengths of each wave and relation between them.

 Example: Using the data given in the picture below; find the wavelength, velocity and amplitude of the wave. Frequency of the source is 2s-1.

We find wavelength of the wave from the picture as;

8m

We find velocity of the wave by using the following formula;

v=wavelength.freguency=8m.2s-1=16m/s

Amplitude of the wave is 2m from the given picture.

Interference of Spring Waves

When two waves interfere they produce resultant wave. In this section we learn how to find resultant wave. Displacement of the resultant wave is the sum of the waves producing it. Look at the given pictures below. They show the behavior of the waves before the interference, and after the interference.

Example: Amplitudes of the waves given below are A1 and A2. Picture shows  the interference of two waves.

Picture given below shows two identical waves interference having opposite directions.

Be Careful!

If the amplitudes of the waves having opposite directions are different, then amplitude of the resultant wave becomes the difference of the amplitudes of waves. It has the same direction with the bigger wave.

Direction of Wave Propagation

If we know the shape of the pulse at an instant time or propagation direction of the particles of the pulse we can find the direction of wave propagation.

Example: Given picture below shows the direction of wave propagation. Find the directions of the vibration at points A, B and C.

We draw the shape of the pulse after t s and find the directions of the vibration at points A, B and C.

Velocity of the Spring Pulse

Distance taken at an instant time by the pulse is called velocity of the pulse. Velocity of the spring pulse depends on the force exerted on the spring and spring constant µ. Spring constant depends on the type of the spring; it is found by the following formula;

 µ=mass/length

Where; v velocity of the spring pulse (m/s),

µ is the spring constant(kg/m), and F is the force exerted on the spring (N).

Example: Find relation of the propagation velocity of pulses of identical springs in given picture below.

Force exerted on the springs directly proportional to the hanged masses. Thus, since G3>G2>G1, 

v3>v2>v1.

Example:   There are three identical springs having equal masses and different lengths L3>L2>L1. Find the relation of velocities of pulses.

Propagation velocity of the pulse is;

Spring constant is;

 µ=mass/length

Relation of the lengths,

L3>L2>L1

Relation of the spring constants;

µ3<µ2<µ1

Propagation velocities become;

v3>v2>v1