Signals/waves can be viewed as objects in what is referred to as a vector space, and such a space is equipped with two very important operations involving its objects: objects can be added together resulting in which is referred to as linear superposition, and an object can be multiplied by a number, which, in the context of sound waves would correspond to changing a sound’s volume. When modeling what is happening mathematically, we are led one of the fundamental algebraic structures in mathematics, namely, that of a vector space. When sound waves are combined, the results can be quite complicated, yet, our ears are able to disentangle some sound components and hear them as separate units. This is a stereo recording, so there are two plots displayed, one for each channel. So two sinusoids at different phases end up producing the effect of a single sinusoid.įor example, here are two sinusoids at the same frequency but with different amplitudes and phases. We can use some standard trigonometric identities to write this asįor some appropriate choice of \(A\) and \(\phi\). More generally, what happens when we play two sinusoids of given amplitudes and phases but the same frequency simultaneously? When we combine the sinusoid \(A_1 \sin(2\pi (ft \phi_1))\) and \(A_2 \sin(2\pi (ft \phi_2))\) to produceĪ_1 \cos(2\pi (ft \phi_1)) A_2 \cos(2\pi (ft \phi_2)) So we see that it is possible for two sinusoids with the same frequency and different amplitudes and at different phases can combine to form a single sinusoid at the same frequency with some new amplitude and phase. Using basic trigonometric identities, the basic sinusoid above can be expressed as a superposition of two different sinusoidsĪ \sin(2\pi (ft \phi)) = A_1 \sin(2\pi ft) A_2\sin(2\pi (ft 1/4)) We can create the sound of a sinusoid with a given amplitude and frequency using a synthesizer and when we have two synthesizers we playing together, the result is the sum of two function formed by summing two functions. We can represent the \(x\)-coordinate of the position at any future time \(t\) by the formula \(\cos(2\pi ft).\) On the other hand, the formula \(\sin(2\pi ft)\) defines the \(y\)-coordinate of the position at a future time \(t\) which is the \(x\)-coordinate phase-shifted by a quarter of a cycle i.e. moves a distance \(2\pi f\) per second), then in Cartesian coordinates, the position at time \(t\) is given by #Soundwaves are full#If our point starts at \((1,0)\) at time \(t=0\) and moves at a speed of \(f\) full cycles of the circle per time unit (i.e. We assume our circle has a radius of 1 unit, making the circumference \(2\pi\). Assuming that the point has moved by an angle \(\theta\) from the point \((1,0)\) on the \(x\)-axis, we call its \(y\)-coordinate the sine of the angle \(\theta\), denoted by \(\sin(\theta)\) and we call its \(x\)-coordinate the cosine of \(\theta\), denoted by \(\cos(\theta).\) The result is shown in the bottom portion of the figure. At each moment in time, the Cartesian coordinates \((x,y)\) of the point can be recorded, and we can plot either the \(x\) or \(y\) coordinate as a function of time. This quantity is referred to as the sinusoid’s frequency. Our findings suggest that sound waves are potential agricultural tools for improving crop growth performance.The speed at which the point rotates about the orign can be measured in terms of the number of complete cycles made per second. Additionally, the cytokinin and auxin concentrations of the roots of Arabidopsis plants increased and decreased, respectively, after exposure to sound waves. Analysis of the expression levels of genes regulating cytokinin and auxin biosynthesis and signaling showed that cytokinin and ethylene signaling genes were downregulated, while auxin signaling and biosynthesis genes were upregulated in Arabidopsis exposed to sound waves. Root development was affected by the concentration and activity of some phytohormones, including cytokinin and auxin. Furthermore, genes involved in cell division were upregulated in seedlings exposed to sound waves. The root length and cell number in the root apical meristem were significantly affected by sound waves. The results of the study showed that Arabidopsis seeds exposed to sound waves (100 and 100 9k Hz) for 15 h per day for 3 day had significantly longer root growth than that in the control group. Therefore, the aim of this study was to examine the effect of sound waves on Arabidopsis thaliana growth. However, the mechanisms by which plants respond to sound waves are largely unknown. Sound waves affect plants at the biochemical, physical, and genetic levels.
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