Greetings again my dear friends I hope you are all very well, so far we have related to various examples, where we involve the phenomenon of sound and some of its intrinsic phenomena at the time of its propagation, such as the Doppler effect and the determination of each of the variables involved in the general formulation of this phenomenon, likewise we relate to exercises related to the reflection or echo of sound.

Now we will analyze in a general way another intrinsic phenomenon, as it is, the phenomenon of refraction of sound waves, and for this to occur, a sound wave front must pass or propagate from one material or elastic medium to another, This will cause this sound wave front to experience changes in its directionality and, of course, in its propagation speed, noting that it is possible that sound waves can be refracted in the same material medium, which is not homogeneous, for example, the air, since it has different density and temperature.

However, this time I wanted to focus on that refraction of sound propagating through two different material or elastic media, such as air and water, so the sound waves or mechanical waves will experience variety in their propagation speed, remember that under standard conditions (20 ° C) the speed of sound in the air oscillates around 340 m / s, while in fresh water is about 1. 435 m/s, that is to say, 4.2 times faster the propagation of sound in water, as for the frequency, it remains invariable from one medium to another, because it is a characteristic provided by the emitting focus of these sound waves.

Therefore, in order to continue deepening on the intrinsic phenomena of sound, in this case its refraction, we will analyze another example, through an exercise like the one you can see below.

Exercise

A beginner trumpet player decides to perform his trumpet manipulation exercises far from his home, in order not to damage anyone's hearing, while he learns to give a harmonic use to this brass wind instrument, he performs his practices near a small lake, He does not realize that one of his friends followed him, and went into the water in order to witness those sharp sounds, so he is not wearing the full diving suit, leaving his ears uncovered, taking advantage that the water depth is not so deep, and the pressure for his ears is not harmful.

Once in the depth of the water he begins to perceive those high-pitched sounds emitted by his friend, the distance of the source of the sound (trumpet), is located at an oblique position about 3.5 meters from the water surface, and also, his friend (the diver) is 10 meters away from the source of the sound emission, as shown in the figure at the beginning of the article, so that the sound waves can be captured by the diver, they have to pass through two material media such as air and water, where, the sound has different speeds, for air 340 m/s, and in fresh water 1. 435.m/s, in relation to the above mentioned, and knowing that the frequency of these sound waves is 1.250 Hz, keeping this magnitude invariable for being characteristic of the musical instrument, answer the following questions:

a.-What is the time implemented by the sound of the trumpet to reach the ears of your friend who is in the water?

b.-What is the value of their wavelengths, i.e., wavelength in air and wavelength in water?

Solution

Data

fo = 1,250 Hz (Frequency of sound from the emitting source).
do = 3.5 m (Distance traveled by the sound waves to the water surface).
da = 6.5 m (Distance traveled by the sound waves from the water surface to the receiver of the sound waves).
Dt = 10 m (Total distance traveled by the sound waves as they are emitted until they reach the diver's ears).
Vaire = 340 m/s (Velocity of the sound propagation medium in the air).
Vagua = 1.435 m/s (Velocity of the sound propagation medium in fresh water).
taire = ? (Time of sound waves in air).
tagua = ? (Time of sound waves in water).
T = ? (Total travel time of sound to reach the diver's ears).
ʎair = ?(Wavelength in air).
ʎwater = ?(Wavelength in water).

a.- To begin to solve this interesting question, we can start by pointing out the data we have, in this case, the velocities of propagation in the two material or elastic media (air and water), but we also have the distances traveled by the sound waves from the precise moment they were emitted, therefore, we can implement the following formula 1.

From the above formula 1, we clear the variable to be searched, i.e. time (t).

We will implement this formula 2 to find two time periods, that is, the time it takes for the sound waves to reach the surface of the water, and the second time, the time that elapses from the surface of the water to reach the ears of the diver who is in the water, therefore, let's start with the time of the sound waves propagating in the air.

Now we do the same for water, adapted to the conditions for that material medium, as you can see.

With these two particular moments or times (taire and tagua) we proceed to find the total time (T) of propagation of the sound waves from the moment they were emitted until they reached the ears of the diver friend of the trumpeter, as you can see.

This was the total time taken by the sound waves to reach the person submerged in the water, highlighting that although the distance traveled in the water was greater than the distance traveled in the air, these waves used less time in the water than in the air, thus proving that sound is refracted when passing from one material medium to another, changing direction and speed.

b.- To give an answer to this unknown related to the wavelengths in each material medium, we will use the following formula.

From this formula we will find the wavelength when the sound propagated through the air.

This is the value of the wavelength while the sound emitted by the trumpet propagated through the air, let's see the wavelength in water.

In this way we observe how both the velocities of propagation of sound and its wavelength change when passing from one medium to another, generating the well-known phenomenon of refraction.

Analysis of results

We continue analyzing the different intrinsic phenomena of sound, in this case, the phenomenon of refraction, which occurs at any place and time where we are, that is, space-time, this time we noticed the change in the direction of sound waves when they pass from one material medium to another, in this case air-water, this makes the speed of these waves vary, and also vary the wavelength.

We also analyzed the time variable implemented by these sound waves from the moment they are emitted until they reach the diver's ears, called total time (T), which is composed of the addition of the particular times of the two material media used for their propagation, observing how upon reaching the water, the refraction of sound causes the change in speed, and although the distance traveled in the water is greater, the time was much shorter.

Undoubtedly, there are many applications of the phenomenon of sound and, above all, of each of these intrinsic phenomena, which occur at the time of the propagation of a certain acoustic signal from one side to another, later we will continue to know other practical examples analyzed from a mathematical vision through various formulations, simple, but very useful when analyzing this type of learning.

Until another opportunity my dear friends.

Note: The images were made by the author using Power Point and Paint, the animated gif was made using PhotoScape.

Recommended Bibliographic References

[1] MOVIMIENTO ONDULATORIO. Link.

[2] REFLEXIÓN Y REFRACCIÓN DEL SONIDO. Link.

[3] SPEED OF SOUND. Link.

[4] Echo-based measurement of the speed of sound. Link.

[5] Propiedades de las ondas sonoras . Link.