Sound Waves and Wave Motion

Fundamental Concepts of Oscillatory Motion

In the study of physics, specifically oscillatory motion, the magnitude of the maximum displacement of a body to one side from its equilibrium position is defined as the amplitude. The symbol used to represent amplitude is $a$, and its SI unit is the metre ($m$). An oscillation is considered complete when a vibrating body returns to its initial position while moving in the same direction from which it started. For example, if a pendulum starts from an equilibrium position $O$, it completes one oscillation by reaching point $A$, traveling to point $B$ on the opposite side, and returning to $O$. Alternatively, if the count begins at point $A$, one oscillation is completed when the pendulum reaches $B$ and returns to $A$.

Historically, pendulum clocks such as the Grandfather's clock utilized a seconds pendulum with a length of approximately $99.35\,cm$ (about $1\,m$). These pendulums were often constructed from an alloy known as invar, which stands for "invariable," due to its stability. In practical applications, if a pendulum takes $1\,minute$ ($60\,s$) to complete $30$ oscillations, the time for one single oscillation is calculated as $\frac{60\,s}{30} = 2\,s$. This time taken for one oscillation is formally called the period, denoted by the symbol $T$, with the SI unit of second ($s$). Conversely, the number of oscillations completed in one second is defined as the frequency, denoted by the letter $f$. Using the previous example, the frequency would be $\frac{30}{60} = 0.5\,Hz$. The SI unit of frequency is the hertz ($Hz$), named in honor of Heinrich Rudolf Hertz.

Heinrich Rudolf Hertz and Frequency Units

Heinrich Rudolf Hertz ($1857-1894$), born in Hamburg, Germany, made significant contributions to physics by experimentally proving the existence of electromagnetic waves. His work laid the foundational principles for future advancements in radio, telephone, telegraph, and television technologies. He is also credited with the discovery of the photoelectric effect. Because of his profound impact on the field, the unit of frequency was named the hertz. In modern telecommunications and electronics, practical units of frequency include the kilohertz and megahertz. One kilohertz ($1\,kHz$) is equal to $1000\,Hz$ or $10^3\,Hz$, while one megahertz ($1\,MHz$) is equal to $1,000,000\,Hz$ or $10^6\,Hz$.

Natural Frequency and Factors of Influence

Every object has a natural frequency, which is the innate frequency at which it vibrates when allowed to move freely. Several physical factors influence this natural frequency: the length of the object, its size, its elasticity, and the nature of the material from which it is constructed. Any change to these factors will result in a corresponding change in the object's natural frequency. For instance, in a simple pendulum, increasing the length of the pendulum leads to a decrease in its frequency. The relationship between period and frequency is inverse; as the period ($T$) increases, the frequency ($f$) decreases, expressed by the formula $f = \frac{1}{T}$. Tuning forks are common tools used in sound experiments, and the markings found on them indicate their specific natural frequency, such as $256\,Hz$.

Forced Vibration and Resonance

Forced vibration occurs when an object is induced to vibrate by an external vibrating source. A common example is the vibration of a table when a mechanical device, such as a mixer or an excited tuning fork, is placed upon it. If a tuning fork's stem is pressed against a table, the sound becomes louder because the table is forced to vibrate at the frequency of the tuning fork. Resonance is a specific condition of forced vibration that occurs when the frequency of the external forcing object matches the natural frequency of the forced object. Under resonance, the object vibrates with its maximum possible amplitude.

Applications of forced vibration and resonance are found in various fields. In medicine, MRI (Magnetic Resonance Imaging) scanning and the use of stethoscopes depend on these principles. A stethoscope allows medical professionals to hear faint heartbeats by utilizing forced vibration and resonance to amplify sound. In communication, radio tuning relies on matching frequencies. Musical instruments like the guitar, violin, veena, harmonium, and mridangam use resonance to produce rich, loud tones. Other examples include megaphones, horns, and wind instruments such as trumpets and nagaswaram.

Wave Motion and Energy Transfer

Wave motion is a primary mode of energy transfer from one part of a medium to another through continuous oscillations, without the actual displacement of the medium's particles over long distances. Energy received in one part of a medium is passed to adjacent parts. Waves can be categorized based on their requirement for a medium. Mechanical waves, such as sound waves, seismic waves, and water ripples, require a medium (solid, liquid, or gas) for transmission. Electromagnetic waves do not require a medium and can travel through a vacuum. Examples of electromagnetic waves include radio waves, microwaves, infrared rays, visible light, ultraviolet rays, X-rays, and gamma rays.

Longitudinal and Transverse Waves

Mechanical waves are further classified into two types: longitudinal and transverse. Longitudinal waves are those in which the particles of the medium vibrate parallel to the direction of the wave's propagation. Sound is a classic example of a longitudinal wave. It travels through air by creating alternating regions of high pressure, called compressions ($C$), and low pressure, called rarefactions ($R$). In a compression, the distance between molecules decreases, whereas in a rarefaction, it increases.

Transverse waves are characterized by particles vibrating perpendicular to the direction of wave propagation. In these waves, the points of maximum elevation from the equilibrium position are called crests, and the points of maximum depression are called troughs. Examples include waves on a plucked string or electromagnetic waves. While longitudinal waves involve pressure variations in the medium, transverse waves (in many contexts) do not. A slinky can demonstrate both: moving it back and forth parallel to its length creates longitudinal pulses, while moving it up and down perpendicular to its length creates transverse waves.

Characteristics of Waves and Mathematical Relationships

The primary characteristics used to describe waves are amplitude, period, frequency, wavelength, and speed. Wavelength, denoted by the Greek letter $\lambda$ (lambda), is the distance between two consecutive particles that are in the same phase of vibration. In transverse waves, this is the distance between two consecutive crests or troughs; in longitudinal waves, it is the distance between two consecutive compressions or rarefactions. The unit of wavelength is the metre ($m$).

The speed of a wave ($v$) is the distance it travels in one second, measured in $m/s$. There is a definitive relationship between these properties: the speed of a wave is the product of its frequency and its wavelength, represented by the equation $v = f \times \lambda$. If the speed of a wave remains constant, the frequency is inversely proportional to the wavelength. For example, if a longitudinal wave travels at $350\,m/s$ with a frequency of $35\,Hz$, its wavelength (and thus the distance between compressions) is $\frac{350}{35} = 10\,m$.

Reflection of Sound, Echo, and Reverberation

Sound waves reflect when they encounter surfaces. Smooth surfaces reflect sound more effectively than rough surfaces, a principle utilized in the design of soundboards and curved ceilings in large halls to distribute sound evenly. An echo is the distinct repetition of a sound heard after the initial sound due to reflection. This is possible because of the "persistence of hearing," a phenomenon where the auditory experience of a sound lasts for approximately $\frac{1}{10}$ of a second ($0.1\,s$) in the human ear. If a reflected sound reaches the ear after this time interval, it is perceived as a separate echo.

To hear a distinct echo in air where the speed of sound is roughly $350\,m/s$, the sound must travel a total distance of at least $35\,m$ ($350\,m/s \times 0.1\,s$). Therefore, the reflecting surface must be at least half that distance, or $17.5\,m$, away from the source. In water, where sound travels at approximately $1480\,m/s$, the minimum distance for an echo would be significantly higher. Reverberation, distinct from a single echo, is the lingering or "boom" of sound caused by multiple reflections in an enclosed space, such as the whispering gallery of Gol Gumbaz in Bijapur, Karnataka. To reduce unwanted reverberation, the walls of cinema theaters are often made rough to absorb rather than reflect sound.

Limits of Human Audibility and Ultrasonic Applications

Human hearing is limited to a specific range of frequencies. For a normal person, the lower limit is approximately $20\,Hz$ and the upper limit is approximately $20,000\,Hz$ ($20\,kHz$). Sounds with frequencies below $20\,Hz$ are termed infrasonic (e.g., seismic waves), while those above $20,000\,Hz$ are termed ultrasonic. While humans cannot hear ultrasonic frequencies, many animals can. For example, Galton whistles used for dog training produce sounds at about $30,000\,Hz$. Bats use ultrasonic waves to navigate and hunt in the dark.

Ultrasonic waves have numerous practical applications, particularly in medicine and industry. In the medical field, they are used for ultrasonography to visualize internal organs like the kidney, liver, and uterus, as well as for treatments like crushing kidney stones or in physiotherapy. In industry, they are used to clean irregular machine parts and electronic components. Furthermore, SONAR (Sound Navigation and Ranging) technology uses ultrasonic waves to detect underwater objects. If a SONAR signal reflects off a rock and returns in $0.2\,s$ through seawater (speed $1522\,m/s$), the distance to the rock is calculated as $d = \frac{1522\,m/s \times 0.2\,s}{2} = 152.2\,m$.

Seismic Waves and Destructive Waves

Seismic waves are waves that travel through the Earth's crust, typically resulting from earthquakes, volcanic eruptions, or massive explosions. Seismology is the specialized study of these waves. The intensity of an earthquake is measured using the Richter scale. When earthquakes occur on the ocean floor or near coastlines, they can displace massive volumes of water, triggering a series of gigantic waves known as a tsunami. Tsunami warning centers provide critical instructions and information to safeguard coastal populations from these destructive events.

Questions & Discussion

• Question: what is the maximum displacement to one side from the equilibrium position? Response: This magnitude is known as the amplitude ($a$).
• Question: When does the swing complete one oscillation? Response: A swing completes one oscillation when the pendulum starts from O, goes to both sides (A and B), and then returns to O. Alternatively, if starting from A, it reaches B and returns to A.
• Question: What could be the reason that the number of oscillations counted by the child waiting for his turn to swing and the child on the swing were different? Response: This difference is subject to discussion regarding observation points and counting accuracy; however, understanding the formal definition of an oscillation (returning to start point in the same direction) ensures accurate counting.
• Question: If a pendulum takes 1 minute to complete 30 oscillations, how long does it take to complete one oscillation? Response: Time for 30 oscillations = $60\,s$. Time for 1 oscillation ($T$) = $\frac{60}{30} = 2\,s$.
• Question: Find the number of oscillations the same pendulum completes in one second. Response: Number of oscillations in 1 second ($f$) = $\frac{30}{60} = 0.5\,Hz$.
• Question: What is the change in frequency when the length of the pendulum increases? Response: When the length increases, the frequency decreases.
• Question: What is the relation between the marking on the tuning fork and its number of vibrations? Response: The marking (e.g., $256\,Hz$) indicates the natural frequency, which is the number of vibrations it makes in one second.
• Question: What is the change in the sound heard when the stem of an excited tuning fork is pressed on the table? Response: The sound becomes louder due to the table undergoing forced vibration.
• Question: When blade A vibrates why would the hacksaw blades C and E vibrate with maximum amplitude? Response: Because the natural frequency of C and E are equal to the natural frequency of A, causing resonance.
• Question: The frequency of a simple pendulum is 1 Hz. What is its period? Response: $T = \frac{1}{f} = \frac{1}{1} = 1\,s$.
• Question: If a pendulum takes 0.5 s to complete one oscillation, what is its frequency? Response: $f = \frac{1}{0.5} = 2\,Hz$.
• Question: Does sound from a source always travel directly to the listener in a hall? Response: No, sound can reach the listener after reflecting off walls, ceilings, and other surfaces.
• Question: Why don't we hear echo inside a small room? Response: Because the reflecting surfaces are closer than $17.5\,m$, so the reflected sound returns within $0.1\,s$ and is merged with the original sound due to persistence of hearing.
• Question: If the frequency of a longitudinal wave travelling at 350 m/s is 35 Hz, what is the distance between two consecutive compressions? Response: The distance is the wavelength $\lambda = \frac{v}{f} = \frac{350}{35} = 10\,m$. The distance between rarefactions is also $10\,m$.
• Question: A sound wave with frequency of 175 Hz has a wavelength of 2 m. Calculate the speed. Response: $v = f \times \lambda = 175\,Hz \times 2\,m = 350\,m/s$.