WAVES AT BOUNDARIES

WAVES AT BOUNDARIES• A wave transmits energy

through a medium

• Eventually that wave will reach a boundary

• A boundary is an obstacle or even another medium that the wave reaches as it travels

WAVES AT BOUNDARIES

• What happens at the wave boundary?

1. The initial wave travels at the boundary. This is the incident wave.

2. A portion of the energy flows into the new medium. This is called the transmitted wave.

3. A portion of the energy is reflected. This is called the reflected wave.

WAVES AT BOUNDARIES

• The incident wave and reflected wave can have 2 situations:1. Fixed End2. Free End

• The transmitted wave depends on the density of the new medium

FIXED END REFLECTION

• Fixed end reflection is when one end of the medium is attached to an object that cannot move

• Think of a rope attached to a table

Let’s talk about sending a wave pulse through this rope to help you visualize

• The incident wave travels through the medium and hits the boundary

• This will created a transmitted and wave and reflected wave

• Let’s think about what will happen to the reflected wave…

FIXED END REFLECTION• The rope is pulling up at the fixed end.

• For every reaction there is an equal and opposite reaction• The fixed end pulls down on the rope

• The reflected wave comes back through the rope but has an inverted amplitude

• The wavelength of the reflected wave is the same as the wavelength of the incident wave

• The speed of the wave is still the same. Remember, speed depends on wavelength and frequency. These do not change.

• The amplitude of the reflected wave is less than the amplitude of the incident wave• Remember, the amplitude squared of a mechanical wave is proportional to the

energy of the wave. Some of the energy was sent to create a transmitted wave.

FREE END REFLECTION

• When the end of a medium is not attached and is free to move, we consider this the case of free-end reflection

• Once again, let’s think about what happens when a wave pulse reaches the end of the rope

• It will create a reflected and transmitted wave again.

FREE END REFLECTION• The rope pulls up at the free end but nothing pulls down.

• The reflected wave returns with but does not have an inverted amplitude

• The wavelength remains the same for the reflected pulse

• The speed of the wave remains the same

• The amplitude is not inversed and is less than the incident wave

TRANSMITTED WAVE

• No we will look at the transmitted wave at a boundary

• The transmitted wave does not depend on free or fixed end

• The transmitted wave depends only on the density of the medium for the incident wave and the density of the medium for the transmitted wave

• The transmitted wave can be in either:1. Higher density medium2. Lower density medium

HIGH DENSITY MEDIUM FOR TRANSMITTED WAVE

• We will examine what happens as a wave passes into a higher density medium from a lower density medium

• As the wave reaches the boundary, some of the energy will continue into the new medium (transmitted wave) and some of the energy will be bounced back (reflected wave)

• The transmitted wave will:• Travel with a slower speed compared to the

incident wave• Have a shorter wavelength than the

incident wave• Amplitude is in the same direction as the

incident wave • Frequency stays the same

• The reflected wave will:• Travel with a same speed as the incident

wave• Have the same wavelength than the

incident wave• Have an inverted amplitude compared to

the incident wave • Frequency stays the same

LOWER DENSITY MEDIUM FOR TRANSMITTED WAVE

• We will examine what happens as a wave passes into a lower density medium from a lower density medium

• The transmitted wave will:• Travel with a faster speed compared to the

incident wave• Have a longer wavelength than the incident

wave• Amplitude is in the same direction as the

incident wave • Frequency stays the same

• The reflected wave will:• Travel with a same speed as the incident

wave• Have the same wavelength than the

incident wave• Amplitude is in the same direction as the

incident wave • Frequency stays the same

FREQUENCY, VELOCITY & AMPLITUDE

• Frequency• The frequency of the wave will be the same in both mediums.

• Velocity• The velocity depends on the properties of each medium and

therefore will be different in each medium. (less dense = faster, more dense = slower).

• Amplitude • The amplitude is directly proportional to the velocity. (if v

higher, amplitude is greater. If v is lower, amplitude is smaller)

CALCULATIONS INVOLVING TRANSMISSION OF WAVES

• From the information we just went through, we should notice that the velocity and amplitude are proportional to each other when a wave changes mediums• If the new medium is less dense, the velocity and amplitude increase• If the new medium is more dense, the velocity and amplitude decrease

• We also learned that frequency remains constant when a wave changes media

• One thing we haven’t discussed yet is what happens to the wavelength when a wave changes media• Recall the “Universal Wave Equation”: v = fλ • If we know that the velocity will change in a new medium, then the

wavelength must also change in a similar way in order for the relationship v = fλ to remain true.

CALCULATIONS INVOLVING TRANSMISSION OF WAVES

• When solving a question where a wave changes mediums, remember that frequency is constant. Solve for it first, then use it to solve for whatever you need.

• Example: A tow rope is connected to a slinky. A wave travelling at 1.50 m/s is introduced to the tow rope. After the wave is transmitted to the slinky, the wave is travelling at 5.35 m/s with a wavelength of 3.00 m. What is the initial wavelength of the wave in the tow rope?• You have enough info to solve for “f” in the slinky

• v = fλ• f = v/λ• f = 5.35 m/s

3.00 mf = 1.78 Hz

• Now, use that value of “f” to solve for the unknown wavelength• v = fλ• Λ = v/f• Λ = 1.50 m/s

1.78 HzΛ = 0.843 m

CALCULATIONS INVOLVING TRANSMISSION OF WAVES

• What we should also realize from this example is the following relationship:

Vi= Vf

λi λf

• Where:• Vi = initial velocity• Λi = initial wavelength• Vf = final velocity• Λf = final wavelength

• From the previous example• Vi = 5.35 m/s • Λi = 3.00 m• Vf = 1.50 m/s • Λf = ?