What is Sinusoidal: Definition and 229 Discussions

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:


A, amplitude, the peak deviation of the function from zero.
f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
ω = 2πf, angular frequency, the rate of change of the function argument in units of radians per second


{\displaystyle \varphi }
, phase, specifies (in radians) where in its cycle the oscillation is at t = 0. When


{\displaystyle \varphi }
is non-zero, the entire waveform appears to be shifted in time by the amount φ/ω seconds. A negative value represents a delay, and a positive value represents an advance.
The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

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  1. Wrichik Basu

    Engineering Sinusoidal steady state analysis using Laplace transform

    ##\require{physics}##The given circuit is this: The question is taken from this video. The Professor has solved it using Phasor analysis, the final solution being $$\begin{equation} i_x(t) = 7.59 \sin \qty( 4t + 108.4^\circ )~\mathrm{amps}. \end{equation}$$My aim, however, is to use Laplace...
  2. C

    Graph for sinusoidal wave travelling to the left

    For part(a) of this problem, The solution is, I don't understand why they assume on the graph where that the waveform is during it's phase. For example, could it not also be correctly drawn as shown in red: Could it not? Many thanks!
  3. C

    Deriving Wave Function for One-Dimensional Sinusoidal Wave

    Where did they get the equation in circled in red from? It does not seem that it can be derived from the graph below. Many thanks
  4. C

    Why Wasn't (0,-3) Included in the Initial Graph of a Traveling Sinusoidal Wave?

    For part(a) of this problem, The solution is, However, why did they not have a point at (x,y) = (0, -3) initially? Also why did they not do a y against time graph?Many thanks!
  5. A

    Sinusoidal sequences with random phases

    Hello all, I have a random sequences question and I am mostly struggling with the last part (e) with deriving the marginal pdf. Any help would be greatly appreciated. My attempt for the other parts a - d is also below, and it would nice if I can get the answers checked to ensure I'm...
  6. Ennio

    B Sinusoidal wave function of t and x

    Greetings, is it possible to characterize a sinusoidal wave in the domain of time and then pass into the domain of movement along x direction? I start with: a is the amplitude of the sine function and ω is the angular velocity. t is the time. I can express the angular velocity in funct. of the...
  7. M

    Mathematica Revolve line around axis in sinusoidal fashion?

    Hi PF! I have a vector valued function ##f(s) = r(s)\hat r + z(s)\hat z## that plots a line in the ##r##,##z## plane when I use ParametricPlot. I'd like to plot this line into a surface, so that it revolves around the ##z## axis, but in a sinusoidal fashion. Basically I'd like to revolve it...
  8. Samama Fahim

    I Time Dependent Sinusoidal Perturbation Energy Conservation

    The transition probability -- the probability that a particle which started out in the state ##\psi_a## will be found, at time ##t##, in the state ##\psi_b## -- is $$P_{a \to b} = \frac{|V_{ab}|}{\hbar^2} \frac{sin^2[(\omega_0 - \omega)t/2]}{(\omega_0 - \omega^2}.$$ (Griffiths, Introduction...
  9. S

    Adding two sinusoidal waves of same frequency but out of phase

    Asin(wt)}+Bsin(wt+a) Asin(wt) +B sin(wt)cos(a) +Bcos(wt)sin(a) Asin(wt) + ksin(wt) + Lcos( wt) (A+K) sin(wt) + Lcos(wt) Fsin(wt) + Lcos(wt)
  10. M

    MHB Finding the Period of a Sinusoidal Model for Ice Cream Sales

    The owner of an ice cream shop kept records of the average number of sales per month for 2019. Create a sinusoidal equation to model this information of number of sales per month.I found the maximum, minimum for this, but how can I find the period of from this table. As I already know formula to...
  11. M

    MHB What Is the Domain and Range of y=cos(3(x - 45°)) +2?

    State the domain and range for one cycle of y=cos(3(x - 45°)) +2 Show your work.
  12. M

    MHB What are the first two positive x-intercepts for the given sinusoidal function?

    Find the first two positive x-intercepts for y= -2cos(3(x-25°)) +1 (Can someone help me for this)
  13. M

    MHB Sinusoidal Functions (I for this)

    Sinusoidal Functions... Can someone help me with this. Describe the transformations that are applied to y= -4cos[2(x-30°)] +5 (State any shifts, stretches, compressions, or reflections).
  14. J

    Why is the sinusoidal considered the fundamental frequency?

    What property of a sinusoid makes it so special? I understand Fourier analysis, but really you could do Fourier using any periodic function as the building block. Sinusoids really do seem to be fundamental though, if you narrow the pass band of a filter with any random signal you will get a...
  15. MisterH

    A Fitness function for window length of filter

    Fitness function for window length of filter On a sinusoidal signal with amplitude 1, and period P, an exponential moving average (EMA) (with alpha = 1/n), and a linear weighted moving average (LWMA) (with window length n) are calculated; when you subtract the EMA from the LWMA, it can be seen...
  16. Rongeet Banerjee

    How can I find the DC component of an Output Sinusoidal Voltage?

    I had previously solved this using Root Mean Square method by integrating the value of voltage from t=0 to t=T/2 and then from t=T/2 to t=T.Answer was Vo/2½.Yesterday I found this question👇🏾 if I followed the previous approach then: 5 volts is not even in the option. How can I find the DC...
  17. mcastillo356

    Solving Sinusoidal Equations with Theta

    I can't solve ##\sin{\theta}\cos{\theta}=0,14## Thanks!
  18. tworitdash

    A Spatial Fourier Transform: Bessel x Sinusoidal

    I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...
  19. PainterGuy

    B Sinusoidal force mechanism for a swing

    Hi, Please have a look on the attachment. The displacement of swing from the equilibrium position, x=0, is considered to be maximum, +x, when the swing reaches the person who is pushing it. The pushing force is of short duration and could be approximated by a pulse. I hope I have it correct...
  20. K

    What is the governing equation of a spring with sinusoidal excitation?

    Hi, Most of the spring vibration lectures assume spring to be fixed on one end and mass on the other end [Example]. In my case, spring has a sinusoidal excitation on one end and mass on other end. Pl. refer the image below. How to get the governing equation? With that I also want to find the...
  21. C

    Class A Amplifier Sinusoidal Input/Output Relation

    My simplistic derivation below for a Class A amplifier shows that an AC signal at the input produces DC, fundamental, and 2nd harmonic terms at the output. This seems to contradict most the information I have found on this - which just states that the output is sinusoidal and of the same...
  22. Specter

    Derivative of a sinusoidal function

    Homework Statement What is the derivative of ##f(x)=\frac {2x^2} {cos x}##? Homework EquationsThe Attempt at a Solution ##F(x)=\frac {2x^2} {cos x}## So... ##f(x)=2x^2## and ##f'(x)=4x## ##g(x)=cosx## and ##g'(x)=-sinx## If I plug these into the quotient rule I thought that I would get...
  23. e0ne199

    Question about sinusoidal steady-state analysis

    hello everyone, i have a problem related with sinusoidal steady-state analysis, the problem is like this : the circuit for the question is like this : i am still unable to find the answer for the question (b), the answer i got is not the same as the one provided by the question above, my...
  24. Talal

    Surface roughness in 2D sinusoidal corrugation

    Hello, For a project I am working on, I am trying to design a 2D with parallel sinusoidal patterns. Imagine a 2D section of a pipe (two parallel surfaces with distance in between). Each surface is basically a sine function. Peak points meet with each other, as well as troughs. Liquid is...
  25. opus

    Graphing sinusoidal tangent functions

    Homework Statement Graph ##y=tan\left(x-\frac {π}{4}\right)## Homework Equations N/A The Attempt at a Solution To graph a tangent function, I first find the vertical asymptotes to set the boundaries for the graph: To do so, set what's inside the parentheses equal to ##\frac π 2## and ##-\frac...
  26. A

    Expressing the addition of two sinusoidal waves this form.

    Homework Statement Express the following in the form x = Re{Aeiαeiωt} (a) x= cos(ωt) + sin(wt) (b) x= sin(ωt +π/4) + cos(ωt) (c) x= 2cos(ωt+π/3) + (√3)sin(ωt)-cos(ωt) Homework Equations cos x = 1/2 e^ix + 1/2 e^-ix sin x = − i/ 2e^ix + i/2 e^−ix The Attempt at a Solution To be honest, I have...
  27. R

    Exploring Oscillatory Behavior of Sinusoidal Functions

    Homework Statement This isn't really part of my homework, my homework was to draw a pretty graph, but I am curious about some behavior. I was given a picture of a sinusoidal function. I found it was ##2sin(\frac{\pi}{3}t-\frac{\pi}{6}) + 6##. Then I used trig identities to get...
  28. karush

    MHB 7.t.27 Write an equation for a sinusoidal graph with the following properties:

    $\tiny{7.t.27}$ $\textsf{Write an equation for a sinusoidal graph with the following properties:}\\$ $$A=-3, \textsf{Period}=\frac{2\pi}{3}, \textsf{Phase Shift}=-\frac{\pi}{4}$$ \begin{align*}\displaystyle A&=-3\\ T&=\frac{2\pi}{3}=\frac{2\pi}{\omega}\\ \omega&=3\\...
  29. B

    I Fourier analysis and the sinusoidal plane wave

    hey So Fourrier transform is ##f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d\omega## with ##F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt## Question 1 - The Fourier mode for the continuous case is ## \frac{1}{2 \pi} F(\omega) e^{i \omega t}##, is...
  30. S

    I Why is the Voltage Induced in a Rotating Coil Sinusoidal?

    Sinusoidal wave form ? I am asking: We know that if a coil rotates in a transverse magnetic field a sinusoidal voltage is induced between its terminals. . My question now is: Why it is exactly sinusoidal in the shape and not any other wave shape?? .
  31. Const@ntine

    Phase Shift of two sinusoidal waves

    Homework Statement Two sinusoidal waves in a string are defined by the wave functions y1 = 2.00 sin (20.0x – 32.0t) y2 = 2.00 sin (25.0x – 40.0t) where x, y1, and y2 are in centimeters and t is in seconds. (a) What is the phase differencebetween these two waves at the point x = 5.00 cm at t...
  32. C

    How to control an ac motor with sinusoidal torque output

    Hello, guys. In field oriented control of ac machines, the electromagnetic torque is proportional to the q-axis current. We can control instantaneous torque by control the q-axis current. If the torque we want is a constant value,with PI controller, no problem, no steady error. But if we want to...
  33. C

    Engineering Calculating Diode Current in R-V Circuit w/ Sinusoidal Input

    Homework Statement R = 1 kohm and Vs(t) is sinusoidal of (peak) amplitude 3 V. The diode is modeled by the series combination of an ideal diode and 0.7 V voltage source. For what percentage of time will the diode conduct? answer: 42.5 Homework EquationsThe Attempt at a Solution I'm...
  34. Captain1024

    Evaluate the Fourier Transform of a Damped Sinusoidal Wave

    Homework Statement Evaluate the Fourier Transform of the damped sinusoidal wave g(t)=e^{-t}sin(2\pi f_ct)u(t) where u(t) is the unit step function. Homework Equations \omega =2\pi f G(f)=\int ^{\infty}_{-\infty} g(t)e^{-j2\pi ft}dt sin(\omega _ct)=\frac{e^{j\omega _ct}-e^{-j\omega _ct}}{2j}...
  35. A

    MHB Finding the zero vector of a sinusoidal set

    Hi everyone, Thank you in advance for taking the time to read my question and for your help. I really appreciate it. The question is set out as follows: Given the set 𝑆 = {𝑎 sin (𝑥+𝑏) | 𝑎, 𝑏 E R}. The addition of elements 𝒔1, 𝒔2 ∈ 𝑆 is defined as follows: 𝒔1 +𝒔2 =𝑎1 sin (𝑥+𝑏1)+𝑎2 sin...
  36. Evangeline101

    Sinusoidal Functions: describe transformations, sketch graph

    Homework Statement Homework Equations none The Attempt at a Solution -amplitude is 3 -period is 180° -right 60° -down 1 Rough sketch of graph: I would like to know if the graph looks right, is there any improvements to be made? Thanks :)
  37. Evangeline101

    Astrolabe Roadstead tides - Sinusoidal Functions

    Homework Statement Homework Equations 3. The Attempt at a Solution a) The height of the high tide is 4.5 m b) The height of the low tide is 0.25 m c) Period = 12.5 hours k= 360/12.5 = 28.8 amplitude = 2.125 m vertical shift = 2.375 m phase shift = it doesn't look like there is any...
  38. S

    I Sinusoidal Potential in Schroedinger

    Hello, How do you solve Schroedinger's equation (time-independent, in one dimension) if the potential is V=sin(x)? Do you have to use the series approximation for sin(x) and obtain a series solution for psi? Is there some way to use Bloch's theorem since the potential is periodic? I've only...
  39. Evangeline101

    Sinusoidal Functions: Niagara Falls Skywheel....

    Homework Statement Homework Equations The Attempt at a Solution a) Here is a sketch of the graph. The lowest point on the Ferries Wheel is 2.5 m and the highest point is 2.5 m + 50.5 m = 53 m. It completes a full cycle every 120 seconds and starts at the lowest point. b) The highest...
  40. T

    Finding amplitude and wave speed in a sinusoidal wave.

    Homework Statement A sinusoidal wave is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in the figure (attachment) for particles at x=0m and x=0.0900m. (A) What is the amplitude of the wave? 4mm (B) What is the...
  41. H

    Expression y(x,t) for sinusoidal wave traveling along a rope

    Homework Statement (a) Write the expression for y as a function of x and t in SI units for a sinusoidal wave traveling along a rope in the negative x direction with the following characteristics: A = 5.00 cm, λ =85.0 cm, f = 5.00 Hz, and y(0, t) = 0 at t = 0. (Use the following as necessary: x...
  42. S

    I Imaginary sinusoidal exponential

    I need to convince myself that ##e^{ikr\text{cos}\ \theta}=\frac{\text{sin}\ \theta}{kr}##. Could you please help me with it?
  43. I

    Calculating Complex Apparent Power in a Sinusoidal Circuit

    Homework Statement In sinusoidal circuit shown in Figure 13 is known : w =10^6 1/s, R = 100 Ohm , L = 300μH , C1 = 10nF and C2= 5nF . Reactive power of coil inductance L is QL = 3kVAr , RMS value of the voltage receiver impedance Z is UZ = 100 V , and the voltage UZ phase delaying behind...
  44. Z

    I Infinite Sq. Well sinusoidal gen. soln question

    I'm reading Griffiths' section on the infinite square well defined as having zero potential between 0 and a on the x-axis and being infinite everywhere else, and am confused about the following part when discussing the general solution inside the well. The bolded part is what confuses me, the...
  45. T

    I Understanding Sinusoidal Waves: Motion and Displacement Explained

    My textbook states the following: The wave disturbance travels from x=0 to some point x to the right of the origin in an amount of time given by x/v, where v is the wave speed. So the motion of point x at time t is the same as the motion x=0 at the earlier time t-(x/v). Hence we find the...
  46. S

    Finding the Intersection of a Sinusoidal Function and a Line

    Homework Statement Hi! I'm trying to find the points of intersection of a sinusoidal function and a line. The line is y=x/7. The function is y=sinx. Can someone tell me how to determine the number of intersections and exact intersections. I would also like to know if the same method can be...
  47. O

    Energy of the summation of two sinusoidal waves

    Suppose we have two laser diodes that are made to transmit light at the same wavelength and intensity. Also, suppose that we place them in an open space and separate them by a distance such that when their generated beams intersect at one point in space and one point only. Further suppose that...
  48. P

    Ionization of hydrogen atom by sinusoidal electric field

    Homework Statement "Suppose that a hydrogen atom, initially in its ground state, is placed in an oscillating electric field ##\mathcal{E}_0 \cos(\omega t) \mathbf{\hat{z}}##, with ##\hbar \omega \gg -13.6\text{eV}##. Calculate the rate of transitions to the continuum." Homework Equations ##R =...