Can there be waves in different fields? I am a physics fan, but I am not a physicist. I don't even know that the question I have asked make sense or not.
There can be waves in the gravitational field. So I would like to know if there can be waves in other types of fields, such as electromagnetic field, Higgs field etc.?
 A: In the electromagnetic field there are photons or light which is a wave. In the Higgs field a Higgs boson can be considered to be an oscillation of the Higgs field (a wave hovering above the ground state of this field). All quantum fields (electron, quark etc) have particles associated with this field and as such these particles are excitations or oscillations in these fields.
But  in the gravitational field, while we have detected gravitational waves, you should know that for this field, the corresponding particle is the graviton which is hypothetical. There is no (satisfactory) quantum field theory for gravity.
A: 
Can there waves in different fields?

Yes. You have already heard about waves in the gravitational field. They were first detected about five years ago!
Waves in the electromagnetic field include light waves, radio waves, microwaves, etc. These electromagnetic waves simply differ in their frequency and wavelength. Otherwise, they are essentially similar because they are all waves in the same EM field and all correspond to the particle called the photon.
Electrons act like waves in an electron field. Quarks act like waves in a quark field. The Higgs particle acts like a wave in the Higgs field.
All quantum particles can act like waves in their respective fields. You can read about wave-particle “duality” here. Physicists discovered that quantum objects can behave like waves or particles, depending on what you do with them!
The Standard Model of particle physics has seventeen different kinds of fields, waves in which correspond to the seventeen known kinds of elementary particles: 6 quarks, 3 charged leptons, 3 neutrinos, 1 photon, 1 gluon, 1 W boson, 1 Z boson, and 1 Higgs. (If you also count different “colors” and antiparticles, you get more than 17, but this is a common way of counting the particles and their fields.)
