Good morning!
Today we're going to build on what we learned last time about standing wave patterns.
As we demonstrated before, standing wave patterns can only be created at specific wavelengths
and therefore specific frequencies.
This demonstrates five different frequencies at which standing wave patterns can be created
on this string.
And again, like last time, I'll add a video echo effect to make it more clear what is
going on here.
♪flipping physics♪
Bobby, what is the equation for the speed of a wave in a medium?
- The speed of a wave equals frequency times wavelength.
- Then Bo, as the frequency of each demonstration increases from 15 to 76 Hz, what must be happening
to the wavelength of the wave on the string?
- Remember, the speed of the wave is going to be constant because the properties of the
medium, the string, remain constant.
- Right.
So that means according to our equation for the speed of the wave, as the frequency increases,
the wavelength of the wave must decrease.
- Correct Bo.
Now let's determine the wavelength of the waves in these standing wave patterns.
Bobby, looking at the first standing wave pattern, the one for 15 Hz, how many wavelengths
are between the two ends of the string?
- I'm sorry but I do not see it.
- OK Bobby, I will admit that the physical demonstrations are great for seeing what the
real physics looks like.
However, sometimes they are less effective at being able to see specifics.
So, I will replace the demonstration with static illustrations of what the waves look like.
Does this help you to be able to see how many wavelengths are in the 15Hz standing wave?
- Yes Mr. P, thanks.
Between the two nodes, there is half a wavelength.
- Great!
Let's define capital L as the length of the string between the two ends.
That means L = 1/2 wavelength.
Therefore, the wavelength of this wave equals 2 times capital L, the length of the string.
- That looks weird.
Why is the 1 in front of the parentheses?
It's not doing anything mathematically.
- Valid point Bo.
I've put it there because there is 1 half wavelength at this frequency and it will help us to identify
a pattern later.
- OK.
- Bo, how many wavelengths in the string for the 30 Hz standing wave pattern?
- It looks like there are two half wave lengths.
That means the wavelength of the wave equals L, the length of the string.
- Correct Bo.
Billy, could you please do the next three.
- Oh yeah, yeah.
So, the 45 Hz standing wave pattern has three half wavelengths, which means the wavelength
equals 2L/3, the 61 Hz standing wave pattern has 4 half wavelengths which means
the wavelength equals L/2.
The 76 Hz standing wave pattern has 5 half wavelengths which means the wavelength equals
2L/5.
- OK, I see what is happening...
The smallest frequency has a 1 in front of the parenthesis and that number increases
by 1 as you get to each of the next possible standing wave patterns.
That's convenient.
- Very good observation Bo.
Now we are going to add some numbers to this and actually solve for the average speed of
the wave on the string in these standing wave patterns.
I measured the length of the string, capital L to be 0.865m.
That means we can create a table and
add all of our frequencies and corresponding wavelengths to it.
We actually determined the equations for the wavelengths and the calculations are relatively
straightforward, so I'm not going to walk through each of those.
Actually, you know what?
Bobby already gave us the equation for the speed of a wave,
it equals frequency times wavelength.
So, calculating the speed of the wave at each frequency is also quite straightforward, we
simply multiply the frequency times the wavelength.
So, I'm also not going to walk through those calculations.
We can then determine the average speed for all five of these frequencies to be 26.1922
or 26 meters per second with two significant digits.
- Yeah.
- OK, so, I know earlier we showed that as the frequency increases, the wavelength decreases
which is exactly what we determined here.
The frequency increases from 15 Hz to 76 Hz and the wavelength of the wave decreases from
1.73m to 0.346m.
However, are you actually telling me all of the standing
wave patterns have roughly the same speed
of 26 meters per second?
I mean, I just - does that make sense?
Could you go back to the real demonstration again?
No, not the one with the weird like reverb effect or whatever it is.
No, the one that shows what is happening at 32 times slower than reality.
Yeah, so this one.
So, you just calculated that all five of those standing wave patterns are all moving at about
the same speed, 26 meters per second.
But look at it, there's no way those five standing wave patterns are all moving at the
same speed, right?
- Yeah.
- Great question Billy, thank you.
Let me explain...
So, when you look at these five standing wave patterns and see that the string in the
76 Hz standing wave pattern is moving up and down faster than the string in the 15 Hz standing
wave pattern.
What you are actually observing is the frequency of the standing wave.
76 Hz or 76 cycles per second is more than five times as many cycles per second as 15 Hz.
So, the 76 Hz standing wave pattern moves up and down more quickly, that is correct.
- Exactly.
- Actually no, the constant 26 meters per second we determined is not the speed at which the
string in the standing wave pattern moves up and down, which remember is not a constant
speed or constant acceleration because the string is moving in simple harmonic motion.
What we determined is the speed of the wave on the string.
Remember the demonstration I did last time of the wave pulse moving back and forth on the string?
- That's right.
OK, standing waves are the constructive and destructive interferences of the wave patterns
going back and forth on the string.
- Yeah, the oscillator on the left keeps sending periodic waves to the right which interfere
with the reflected and inverted waves which moves to the left after running into the fixed
end on the right.
That's what creates the standing wave pattern.
- Remember that whole animation thing he showed us?
The wave speed we calculated in the data table is the speed of the blue and red waves moving
to the left and right through the medium which create the standing wave pattern.
This is confusing.
- Yeah, it is confusing.
- I totally understand that it is confusing, but that is why we are going through it.
Bo, great observation that we are determining the speed of the blue and red waves which
interfere with one another to create the standing wave pattern.
It is difficult to remember because in the real demonstration, we do not actually see
those waves which create the standing wave pattern.
However, if we go back to the single wave pulse I created by plucking the string, realize
we can measure the speed of the wave pulse.
- And it should be the same as the speed we determined using the speed equals frequency
times the wavelength equation.
- Right, all we need is the time it takes for the wave pulse to go there and back again.
- Here it is.
- OK, speed equals distance over time.
The distance is 2 times L, the change in time is 0.068 seconds, so the speed is 25.441 or
25 meters per second with 2 significant digits.
- And the percentage difference between our two ways of calculating the speed of the wave
on the string is 2.910 or 2.9%.
Nice.
- Yeah.
- That's very close.
- I'm impressed...
With the physics.
- Really?
- What!
- Not with us?
- No, absolutely.
I'm impressed by the three of you as well.
However, the physics works, right?
- Uh-huh uh-huh...
- Thank you very much for learning with me today, I enjoyed learning with you.