3.4.3 Experimental principles
Sound is a mechanical disturbance traveling in elastic media(i.e.gases,liquid and solid material).In particular,sound is a longitudinal wave because the particle vibration direction is parallel to the direction of wave propagation.Usually,sound with vibration frequency between 20 Hz to 20 kHz can be heard by human beings.In engineering,a vibration with frequency more than 20 kHz is called ultrasound.At present,ultrasound has been widely used in our daily life,such as various types of sonars,ultrasonic imaging in medicine,flaw detection in industry.
The piezoelectric ceramic ultrasonic transducer has been usually employed to produce ultrasonic waves.Piezoelectricity is a feature of crystals of quartz,barium titanate,Rochelle salt,and other materials,which can be used to transform one type of energy to another one.The piezoelectric ceramic ultrasonic transducer can realize the conversion between sound pressure and electrical field.The schematic diagram of a piezoelectric ceramic ultrasonic transducer is illustrated in Figure 3.4.1.The piezoelectric ceramic ultrasonic transducer is actually a sandwich oscillator composed of a piezoelectric ceramic plate and two metal contacts.The piezoelectric ceramic plate is made of a polycrystalline piezoelectric material,which can exhibit piezoelectric effect after polarization treatment at a certain temperature.The front contact(Radiation head)is made of light metal with a horn shape and the back one(Baffle board)is made of heavy metal with a column shape,which is benefit to increase the radiation area as well as enhance the coupling effect between the oscillator and the medium.
Figure 3.4.1 The schematic diagram of piezoelectric ceramic ultrasonic transducer
As a sound source,the piezoelectric ceramic ultrasonic transducer with the structure mentioned above contains many advantages,such as good flatness,good monochromaticity and strong directionality.In addition,the sound produced by the piezoelectric ceramic ultrasonic transducer will not be disturbed by audio in daily life due to its frequency limitation within the ultrasonic range(＞20 kHz).As is known to all,the wavelength of ultrasound λ becomes shorter as its frequency increases.Therefore,one can observe lots of λ in a short distance in this experiment,which is beneficial to accurately determine the λ by taking its average value.
Sound velocity is a physical quantity to characterize the speed of sound traveling in elastic media.According to wave theory,the velocity and the relationship between frequency and wavelength is v=f λ.Therefore,the velocity v can be calculated by measuring the frequency of acoustic f and wavelength λ.Two kinds of methods,namely the resonant interfering method(standing wave)and phase comparison method,have been widely used to measure the wavelength λ of ultrasonic sound.
1)The resoname interfereme method(standing wave)
(1)Principle of superposition of waves:When two(or more)waves travel through the same medium at the same time,the waves pass through each other without being disturbed.The net displacement of the medium at any point in space or time is the sum of the individual wave displacement.
(2)Standing wave:a standing wave can occur in a stationary medium as a result of the interference between two identical waves with equal amplitude,frequency but traveling in opposite directions.
According to wave theory,set up along the x direction for injection of the incident wave equation:
For the reflected wave equation:
Where A is sound amplitude,ω is angular frequency,and λ is wavelength.
Then,the resultant wave equation will be the sum of y1 and y2:
Equation(3.4.3)is the wave function of a standing wave.Based on Equation(3.4.3),synthesis of vibration amplitude is the most minimum at the nodes but becomes the largest at an antinode locating at midway between two adjacent nodes.Moreover,the distance between adjacent nodes,or adjacent antinodes is one half of the wavelength of the wave,which can be expressed as:d=λ/2.In other words,the wavelength λ of the ultrasonic wave can be calculated once the distance between adjacent nodes,or adjacent antinodes is measured.
Figure 3.4.2 The schematic diagram of resonance interference method(https://www.daowen.com)
In this experiment,the schematic diagram of the resonance interference method is illustrated in Figure 3.4.2.Signal generator outputs a sine AC signal to the transmitter.The transmitter emits an ultrasonic wave with the same frequency,whose wave-front approximates as a plane.This plane sound wave travels over a certain distance to the receiver.The receiver not only receives the transmitted ultrasonic wave(converting the received acoustic signal into electrical signal),but also reflects part of the coming ultrasonic wave.In fact,the surface of the transmitter and receiver will form a resonant cavity because the transmitter and receiver are parallel to each other.Therefore,a standing wave is formed in the resonant cavity due to acoustic interference.Changing the distance between the two transducers,the sound waves in the cavity repeatedly form stable standing wave.When the distance between two transducers equals to the integer times of half-wave λ/2,the receiver will receive the largest sound pressure,and then a maximum electrical signal could be observed in the oscilloscope.
Therefore,in this experiment,we can observe a repeatable maximum amplitude in the oscilloscope displayed at some special points if the frequency of the transmitter keeps constant while changing the distance between transmitter and receiver continuously.The distance between two adjacent maximum amplitude is λ/2.Then,we can calculate the speed of ultrasonic wave v based on the formula v=f λ,where the frequency f is extracted from the signal generator.
2)Phase comparison method
A sound field will be formed around its source once a sound wave emits from the transmitter.The vibration phase of sound field at any point x in the medium will change with time.However,the phase difference between the detection point x and the sound source will not be changed over time.
Assuming that the wave equation at the point of the transmitter is:
Therefore,the wave equation at any point x is:
Comparing the two wave equations above,the phase difference between the transmitter and receiver isΔφ=2πΔL/λ.In other words,the phase differenceΔφ will change 2π when the distance between the transmitter and receiver is changed from L0 to L0+λ,where L0 is the initial distance between the transmitter and receiver.
In this experiment,the drive sine signal of the transmitter and the received sine signal from the receiver are inputted to the X and Y terminal of the oscilloscope,respectively.The synthetic trajectory of two perpendicular vibration with the same frequency is called as Lissajous pattern,as shown in Figure 3.4.3.In one period,the Lissajous patterns undergoes different evolution in turn,including line with positive slope,ellipse,circular,ellipse,and line with negative slope,ellipse,circular,ellipse.Therefore,when the change of the distance between the transmitter and receiver is equal to a wavelength,the phase difference between transmitted and received signals also just changes one period(Δφ=2π)and the same graphic will appear in the Lissajous pattern.Then,we can extract the wavelength based on the distance change in one period.In order to identify one period easily,it is better to choose one line pattern(positive slope or negative slope)as the beginning pattern.
Figure 3.4.3 Lissajous patterns at different phase difference
3)The relationship between the sound speed and the air specific heat ratio
Under the ideal gas state approximation,the travelling process of sound wave can be considered as an adiabatic process.Therefore,the speed of the sound waves in the air can be written as:
Where R is the molar gas constant(R=8.314 J·mol-1·K-1),and γ=Cp/Cv is the specific heat ratio,μ is the molar mass of the gas,and T is the absolute(Kelvin)temperature.
Therefore,using the conversion relationship between Celsius scale and t Kelvin temperature scale T,Equation(3.4.6)can be expressed as:
Where v0=331.45 m/s is the speed of sound in air at 0℃.Therefore,Equation(3.4.7)can be used as a theoretical calculation formula for the speed of sound in air.For example,the speed of sound in air at room temperature(T=293 K)is approximately 343 m/s.
