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3) waves, the classic view / SurprizingFacts

Having dealt with the equations for oscillations – describing almost everything that jumps, vibrates, rolls back and forth, like a ball on a spring – you can go to the equally widespread phenomenon of nature, waves. Waves are everywhere: sound and light, earthquakes, ripples on the surface of a pond, etc.

 image

Fig. 1

But before this I warn that the term "wave" can be misleading, because in physics it means not the same as in English. In physics, it does not mean what we usually could call a wave on the edge of the ocean – one crest and one depression. In physics, waves are a sequence of waves, several crests and valleys that move together in the same direction. At a wave of the simplest kind all ridges of identical height and are spaced from each other on one distance. We will consider just such a case.

Waves are an outstanding phenomenon, if you think about it. Imagine that you and your friend took a long rope and tightly pulled it in the room (Figure 2). Then imagine that your friend chatted up and down several times with one end of the rope (green). A wave will appear at the end of the rope and it will go around the room to your end of the rope (red).

Fig. 2

This is amazing. I mean – it's actually amazing, strong and critical for everything in our Universe, including yourself. Look what happened. No physical object left to right moved – before your friend started moving the end of the rope, it was stretched across the room, and at the end, after your end of the rope has finished hesitating and the wave is gone, the rope will remain stretched across Room, as it was. And still! Energy and information moved around the room. The wave in the way carries the energy spent by your friend on the vibrations of the rope – and carries in its form information about how many times and how quickly he pulled it – to you, where it makes you shake your hand. And in this case, she even shakes your hand so many times and it is in such a sequence. Wow! No physical object moved through the room, and energy and information moved.

Or, wait. Should not we consider the wave as a physical object?

Remembering this deepest question, let us turn to a small number of mathematical formulas necessary for describing the appearance and behavior of a wave, and then use a little more mathematics to write down equations whose solutions are waves . This is similar to what we did for a classical ball on a spring.

A formula for an infinite wave at a certain point in time

This series of articles immediately after the ball on the spring goes to the waves because the wave is a kind of double oscillator. It fluctuates both in time and in space. We denote the time by the letter "t", and the space by "x".

Notice in Fig. 3. It depicts a wave extending in both directions over a large distance, on which many ridges and valleys fit. This differs from the wave in Fig. 2, which has only a few crests and depressions. But this difference is irrelevant – in Fig. 2 I needed to illustrate something for which the exact waveform did not matter; Now we will concentrate on the mathematical formula for waves, and this is much easier to do if the wave has a large number of ridges and depressions of the same size. Also this case will be very useful for understanding how quantum mechanics affects the behavior of waves.



Fig. 3

First we need to determine the notation and write down a formula describing the motion and waveform in Fig. 3, as we did for the ball on the spring.

The graph shows the magnitude of the wave Z as a function of space in a certain time period t = t 0 – we write this as Z (x, T 0 ). By tracking the wave in space, we see that it oscillates back and forth, and Z periodically increases and decreases. At any given time, the wave oscillates in space.

Note that Z does not need to be related to physical distance. This can be the height of the rope, as in Fig. 2, or it could be something completely different, for example, the air temperature at a certain point in space and time, or the orientation of the magnetic atom in a certain place of the magnet. But x still represents the physical distance, and t is the time.

The image of this wave, Z (x, t 0 ), has three interesting properties, two of which also apply to Ball on the spring.

1. There is an equilibrium value Z 0 lying in the middle between the largest value of Z on the crest and the smallest value of Z in the cavity. Most of the time we study waves in which Z 0 = 0, since often the value of Z 0 does not matter – but not always.
2. The wave has an amplitude A, a value by which Z varies from the equilibrium value to the top of each crest or by the same amount to the bottom of each depression.
3. A wave has a length – a distance λ between adjacent ridges, or, which is the same, between adjacent depressions, or, equivalently, twice the distance between adjacent ridge and depression. It describes oscillations in space in the same way as the period (equal to 1 / frequency) describes the oscillation in time of the ball on the spring.


Fig. 4

What reminds us of the shape in Fig. 3? It looks like a graph of a sine or cosine function – see Fig. 4, where cos (w) is plotted on w. Cos (w) is an oscillating function, which has an obvious equilibrium position at zero, its amplitude is 1, and the wavelength is 2π. How to proceed from Fig. 4 to the formula for the wave in Fig. 3? First we multiply cos (w) by A so that the amplitude is equal to A. Then we add Z 0 to the whole formula to shift it to the desired equilibrium value (if A = 0, then there is no oscillation, and everything Rests at the point Z = Z 0 ). Finally, we replace w by 2πx / λ, since cos (w) has crests at w = 0 and w = 2 π, so cos (2πx / λ) has ridges at x = 0 and x = λ. All this gives us

 $ Z (x, t_0) = Z_0 + A cos (2π x / λ) $ "data-tex =" display

This is practically the same formula that described the motion of a sphere on a spring in time:

 $ z (t) = z_0 + A cos (2π ν t) = z_0 + A cos (2π t / T) $ "Data-tex =" display

Where ν is the oscillation frequency, and T = 1 / ν is the oscillation period. You see the analogy: the period refers to time as the wavelength to space.

One more remark before we continue. I could also write:

 $ Z (x, t_0) = Z_0 + A cos (-2πx / λ) $ "data-tex =" display

Because cos [w] = cos [-w]. The fact that we can easily substitute a minus in the waveform formula will be important later.

The formula for an infinite wave in a certain place


Fig. 5

Now let's ask another question: let's see how the wave changes in time, tracing a certain point on the rope, and see how it behaves and moves. This is shown in Fig. 5: there I designated a certain point x 0 which at the time t 0 is on the crest. The wave moves to the right and follows the wave size Z at the point x 0 varying in time: Z (x 0 t). And you will immediately see that the height of the wave at a certain point behaves exactly like a ball on a spring! Therefore, it will have exactly the same formula as a ball on a spring, as a function of the frequency ν, or of the period T = 1 / ν, where T is the time between the moment when the wave at x 0 The crest, and the moment when it again approaches the ridge the next time.

 $ Z (x_0, t) = Z_0 + A cos (2π ν t) = Z_0 + A cos ( The complete formula of an infinite wave </h2>
<p> The total formula of an infinite wave </h2>
<p>
Now we need a formula for Z (x, t) describing the wave shown in Fig. 3 and 5 (or any similar) at the points x at any time t. The correct answer is: </p>
<p><math> $$ display $$ Z (x, t) = Z_0 + A cos (2π [ν t – x/λ]) = Z_0 + A cos (2π [t/T – x/λ]) $$ display $$ </math>
</p>
<p>
It includes both formulas, for a fixed point in time and for a fixed point in space. </p>
<p> Note the minus sign in front of x. I mentioned that in the formula for Z (x, t <sub> 0 </sub>) it is possible to substitute a minus at will. With a minus before x and a plus before t the formula describes a wave moving to the right, as on animations. To verify this, note that when t / T – x / λ = 0, the wave will crest, because cos [0] = 1. When t = 0, at the point x = 0 the crest. But if we move t a little forward, say, T / 10, then the comb will be at the point x = λ / 10, to the right of the place where it was at t = 0 – therefore the crest (and the whole wave) moves to the right. </p>
<p> ] </p>
<p> What will change if you place plus instead of minus in the formula for Z (x, t)? Then the comb will be at the point t / T + x / λ = 0, and in this case at t = T / 10 the comb will be at the point x = -λ / 10, to the left of the place where it was at t = 0 – means , Now the wave moves to the left (Figure 6). </p>
<p> <img src=
Fig. 6

Waves that are functions of x and t can move in any direction, so we just need to choose the right formula for a given wave. Generally speaking, when we work with waves that can move not only along one spatial dimension x, but along all three coordinates x, y and z, then these waves can move in any direction, and we need to choose the right formula based on the direction Wave motion.

Small font: we can put a minus sign before t, and not before x. But + t, + x is the same as -t, -x, since this would be equivalent to multiplying the entire formula inside the cosine by -1, and cos [w] = cos [-w]. Therefore, + t, + x and -t, -x give a wave moving to the left, and + t, -x and -t, + x give a wave moving to the right.

Equation of wave motion

Now, as in the case of a ball on a spring, when we first found the formula for the vibrational motion of a ball, and then looked at the equation of motion for which this formula was the solution, we will do the same here. We have found a formula for the shape and motion of a wave. Which equation of motion among the solutions does such a formula occur? We learn in the next article.

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