Practice Problems

Let's discuss why standing waves do not transfer energy. Can anyone explain it?

Is it because the nodes don't move, so there's no energy moving through them?

Exactly! The nodes remain stationary, leading to no net movement of energy along the string. Remember, the positions of maximum displacement, called antinodes, oscillate but are balanced by nodes that do not move.

So, would the energy just reflect back and forth at the nodes?

That's correct, Student_2! The energy is stored in the oscillation between the nodes and antinodes but does not propagate away.

So, is that why they say standing waves are like storing energy?

Great connection! To recap, standing waves don’t transfer energy due to the fixed nodes and balanced oscillations.

Now, let’s dive into impedance and how it influences wave reflection. Who can tell me what impedance is?

Isn't impedance the resistance that waves face when traveling through a medium?

Absolutely! And when two media have different impedances, what happens to the wave?

Part of the wave is reflected, right? And part is transmitted?

Exactly right! The reflection coefficient indicates how much of the wave is reflected back depending on the difference in impedance, Z1 and Z2.

Can we calculate this using the formulas provided?

Yes, you can use R = (Z2 - Z1) / (Z2 + Z1) for reflection and T = (2Z2) / (Z2 + Z1) for transmission. Great job, everyone!

Let’s apply our learning to find the eigenfrequencies for a string fixed at both ends. What is the formula for the eigenfrequencies?

Is it fn=n(v/2L)?

Correct! Now if we have a string of length 1 meter and wave speed of 200 m/s, what would be the first three eigenfrequencies?

For n=1, f1=1(200/21)=100 Hz. For n=2, f2=2(200/21)=200 Hz. And for n=3, f3=3(200/2*1)=300 Hz.

Fantastic job, Student_1! The first three eigenfrequencies are 100 Hz, 200 Hz, and 300 Hz.

So they show us the frequencies at which the string vibrates?

Exactly! These frequencies correspond to the natural vibration modes of the string.

Next up, let’s determine the fundamental frequency for an open-closed pipe of length 0.5 m. What’s our formula?

Is it fn=(2n-1)v/4L for n=1?

You got it! Given that the speed of sound in air is 343 m/s, what is the first harmonic?

Plugging it in: f1=(21-1)(343)/(40.5)=343/2=171.5 Hz.

Great work, Student_3! The fundamental frequency of this pipe is 171.5 Hz.

What does this frequency represent in terms of sound?

This frequency is what we hear as the fundamental pitch of the sound produced by the pipe.

Let’s expand our understanding by discussing phase and group velocities. Can anyone describe the difference?

Phase velocity is the speed of a single wave peak, while group velocity is the speed at which a group of waves travels.

Perfect! How do you calculate phase velocity?

Is it v_p=ω/k?

Exactly! And what about group velocity?

Group velocity is given by v_g=dω/dk.

Well summarized! Understanding both velocities helps us appreciate wave propagation in different contexts. Remember, phase velocity is associated with individual waves, while group velocity relates to energy transfer.