Waves at Boundaries

1. Waves at Boundaries

2. 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

3. Waves at Boundaries • What happens at the wave boundary? • The initial wave travels at the boundary. This is the incident wave. • A portion of the energy flows into the new medium. This is called the transmitted wave. • A portion of the energy is reflected. This is called the reflected wave.

4. Waves at boundaries • The incident wave and reflected wave can have 2 situations: • Fixed End • Free End • The transmitted wave depends on the density of the new medium

5. 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…

6. 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.

7. 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.

8. 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 speedof the wave remains the same • The amplitude is not inversed and is less than the incident wave

9. 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: • Higher density medium • Lower density medium

10. 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 reflected wave will: • Travel with a same speed as the incident wave • Have the same wavelengththan the incident wave • Have an inverted amplitudecompared to the incident wave • Frequency stays the same • 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

11. 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 reflected wave will: • Travel with a same speed as the incident wave • Have the same wavelengththan the incident wave • Amplitude is in the same direction as the incident wave • Frequency stays the same • The transmitted wave will: • Travel with a faster speedcompared to the incident wave • Have a longer wavelengththan the incident wave • Amplitude is in the same direction as the incident wave • Frequency stays the same

12. 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)

13. 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.

14. 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 m f = 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 Hz Λ = 0.843 m

15. Calculations Involving Transmission of Waves • What we should also realize from this example is the following relationship: Vi= Vf λiλf • 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 = ?