WHAT IS THE VALLEY IN PHYSICS? (WITH EXAMPLES) - PHYSICAL - 2025

The valley in physics is a name that is applied in the study of wave phenomena, to indicate the minimum or lowest value of a wave. Thus, a valley is considered as a concavity or depression.

In the case of the circular wave that forms on the surface of the water when a drop or a stone falls, the depressions are the valleys of the wave and the bulges are the ridges.

Figure 1. Valleys and ridges in a circular wave. Source: pixabay

Another example is the wave generated in a taut string, one end of which is made to oscillate vertically, while the other remains fixed. In this case, the wave produced propagates with a certain speed, has a sinusoidal shape and is also made up of valleys and ridges.

The above examples refer to transverse waves, because the valleys and ridges run transverse or perpendicular to the direction of propagation.

However, the same concept can be applied to longitudinal waves such as sound in air, whose oscillations occur in the same direction of propagation. Here the valleys of the wave will be the places where the density of the air is minimum and the peaks where the air is denser or compressed.

Parameters of a wave

The distance between two valleys, or the distance between two ridges, is called the wavelength and is denoted by the Greek letter λ. A single point on a wave changes from being in a valley to being a crest as the oscillation spreads.

Figure 2. Oscillation of a wave. Source: wikimedia commons

The time that passes from a valley-crest-valley, being in a fixed position, is called the period of the oscillation and this time is denoted by a capital t: T.

In the time of a period T the wave advances a wavelength λ, that is why it is said that the speed v with which the wave advances is:

v = λ / T

The separation or vertical distance between the valley and the crest of a wave is twice the amplitude of oscillation, that is, the distance from a valley to the center of the vertical oscillation is the amplitude A of the wave.

Valleys and ridges in a harmonic wave

A wave is harmonic if its shape is described by the sine or cosine mathematical functions. In general, a harmonic wave is written as:

y (x, t) = A cos (k⋅x ± ω⋅t)

In this equation the variable y represents the deviation or displacement with respect to the equilibrium position (y = 0) at position x at time t.

Parameter A is the amplitude of the oscillation, an always positive quantity that represents the deviation from the valley of the wave to the center of oscillation (y = 0). In a harmonic wave, the deviation y, from the valley to the crest, is A / 2.

Wave number

Other parameters that appear in the harmonic wave formula, specifically in the argument of the sine function, are the wave number k and the angular frequency ω.

The wave number k is related to the wavelength λ by the following expression:

k = 2π / λ

Angular frequency

The angular frequency ω is related to the period T by:

ω = 2π / T

Note that ± appears in the argument of the sine function, that is, in some cases the positive sign is applied and in others the negative sign.

If a wave is propagating in the positive x direction, then it is the minus sign (-) that should be applied. Otherwise, that is, in a wave that propagates in the negative direction, the positive sign (+) is applied.

Harmonic wave speed

The speed of propagation of a harmonic wave can be written as a function of angular frequency and wave number as follows:

v = ω / k

It is easy to show that this expression is completely equivalent to the one we gave earlier in terms of wavelength and period.

Valleys example: the clothesline rope

A child plays waves with the rope of a clothesline, for which he unties one end and makes it oscillate with a vertical movement at a rate of 1 oscillation per second.

During this process, the child stays still in the same place and only moves his arm up and down and vice versa.

While the boy generates the waves, his older brother takes a photo of him with his mobile. When you compare the size of the waves with the car parked just behind the rope, you notice that the vertical separation between valleys and ridges is the same as the height of the car windows (44 cm).

In the photo it can also be seen that the separation between two consecutive valleys is the same as that between the rear edge of the rear door and the front edge of the front door (2.6 m).

Harmonic wave function for the string

With these data, the older brother proposes to find the harmonic wave function assuming as the initial moment (t = 0) the moment when his little brother's hand was at the highest point.

It will also assume that the x-axis begins (x = 0) at the hand place, with a positive forward direction and passing through the middle of the vertical oscillation. With this information you can calculate the parameters of the harmonic wave:

The amplitude is half the height from a valley to a ridge, that is:

A = 44cm / 2 = 22cm = 0.22m

The wave number is

k = 2π / (2.6 m) = 2.42 rad / m

As the child raises and lowers his hand in the time of one second then the angular frequency will be

ω = 2π / (1 s) = 6.28 rad / s

In short, the formula for the harmonic wave is

y (x, t) = 0.22m cos (2.42⋅x - 6.28 ⋅t)

The speed of propagation of the wave will be

v = 6.28 rad / s / 2.42 rad / m = 15.2 m / s

Position of the valleys on the rope

The first valley one second after starting the movement of the hand will be at the distance d from the child and given by the following relationship:

y (d, 1s) = -0.22m = 0.22m cos (2.42⋅d - 6.28 ⋅1)

Which means that

cos (2.42⋅d - 6.28) = -1

That is to say

2.42⋅d - 6.28 = -π

2.42⋅d = π

d = 1.3 m (position of the nearest valley at t = 1s)

References

1. Giancoli, D. Physics. Principles with Applications. 6th Edition. Prentice Hall. 80-90
2. Resnick, R. (1999). Physical. Volume 1. Third edition in Spanish. Mexico. Compañía Editorial Continental SA de CV 100-120.
3. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7th. Edition. Mexico. Cengage Learning Editors. 95-100.
4. Strings, standing waves and harmonics. Recovered from: newt.phys.unsw.edu.au

Waves and Mechanical Simple Harmonic Waves. Recovered from: physicskey.com.