Acoustic waves are oscillations of pressure in the atmosphere that travel through solids, liquids or gases. They transmit sound, for example, by vibrating organs in the ear that produce the sensation of hearing. I found the acoustic wave equation which is a mathematical description of acoustic pressure waves. In one dimension, the acoustic wave equation is given by this equation. In this equation, p is the acoustic pressure, x is distance and t is time. I have moderate hearing loss in one ear and from frequently doing hearing tests, I have seen a number of the graphs and am curious as to how they calculate hearing loss. I wonder if technology uses this equation or some variation in the process.
An example of a partial differential equation I found is Laplace equation. This is a second order partial differential equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time and it's often used n multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics (wikipedia).The reason I found it interesting is because I didn't expect that this small formula could be useful in so many fields.
This is an example of the Navier Stokes equations which are used to model the motion of viscous fluids substances. I think this is really interesting because they can be used to model blood flow which is super important for many medical applications including emergency medicine where you are losing blood and treatments involving blood.
Maxwell's equations are a set of partial differential equations that describe the generation of electric and magnetic fields through charges, currents, and changes in fields. The equations have laid a foundation for understanding electromagnetism and electric circuits. They show the rate of change for the electric field, E, and the magnetic field, B, in the presence of a charge, rho, and an electrical current, j.
This equation represents the behavior of waves. Waves on strings or membranes, pressure waves in gasses, liquids or solids, electromagnetic waves and gravitational waves, and the current or potential along a transmission line all satisfy this equation. I found it interesting the behaviors of these different things, which all vary in scale and composition, can all be modeled by one equation.
In economics, you can use partial differential equations to optimize a function subject to a constraint, as seen below (this example is from Khan Academy).
You set up the function and then take partial derivatives with respect to each of the variables. When you solve these out, you get the optimization of the function, which is useful if you are trying to maximize profit with a certain amount of labor and material, for example.
I found this example interesting because I'm an econ major and we did this in my microeconmics class, I was sort of lost the whole time, but now I know what is actually going on when you do this, so thats cool!
These equations are used to study the way tsunamis break onto the coast. They are also used to study the velocity of tsunamis. I found these cool because tsunamis are very interesting in how they form and I think it is important to study them to make sure we can evacuate people in time because tsunamis are often very deadly.
Helmholtz’ equation is a linear partial differential equation that measures the motion of seismic waves through the Earth. Seismic waves are caused by different components of Earth moving, like how a fault may move and cause an earthquake. Volcanoes, explosions, landslides, avalanches, and more are influenced by and cause seismic waves. The Helmholtz’ equation can also be applied in different ways such as the vibrations of a drum being hit or measuring sound waves. I found this equation interesting due to its broad applications, like monitoring Earth’s tremor before a volcano erupts, and how it could help warn residents of an environmental disaster.
This is an example of using partial derivatives in order to change an equation and make it easier to solve with different variables. This is a partial derivative of the gas law equation. I thought this was interesting because I learned about this in chemistry but had never seen partial derivatives used on it in this way and I think it would be interesting to see what this equation could be used for.
The partial differential equation I choose was the Schrödinger equation. This linear partial differential equation helps physicists evalute a wave function to find the relative probabilty of finding the electron at a specific time. If I'm being honest, I don't completely understand it, but neither did Schrödinger.
I choose this equation as I enjoy learning about interpretations of quantum mechanics. I haven't taken physics since high school but for my final presentation I decided to research and rank (based on my opinion) 5 different interpretations of quantum mechanics. I thought it was really cool and interesting.
I also think its funny that the purpose of Schrödinger's thought experiment was to show how quantum superposition wouldn't work at the macroscopic level, but did the opposite and only helped create more thought experiments such as quantum suicide.
I find the equation of marginal product of labor and capital. I chose this because I learned economics this semester. These derivatives are essential in economic decision-making, helping optimize resource allocation for efficient production.
This is Poisson’s equation, a partial differential equation that has broad applications in the realm of theoretical physics. For instance, it can be used to describe the potential field created by an electrical charge, which can allow for the calculation of the electrostatic or gravitational fields. It can also be used in calculations other areas of physics such as newtonian gravity, hydrodynamics, and diffusion. While I don’t have a particularly deep understanding of physics, I think it’s really interesting to see how the mathematical concepts that we’re learning now can be applied to such large conceptual ideas.
These equations show the realationship between predator and prey in a closed model. They assume some things that are not necessarily true in nature but will tell you whether both species will survive, or if either one will be the sole survivor. It also shows how the populations interact. This may not be incredibly accurate for real life animals but for organisms in a laboratory, it can be very useful. I think that this is super interesting as it is much more of a real life application than I normally see for math equations. This is what it looks like. there is a more in-depth explanation as to what the variables mean on the Wikipedia page
A partial differential equation example is the wave equation, which indicates how waves change over time. It can describe various waves such as sound and water waves. I find this interesting as it incorporates the Laplacian operator, representing the wave's spatial variation.